:: ORTSP_1 semantic presentation

reconsider X = {0} as non empty set ;

reconsider o = 0 as Element of X by TARSKI:def 1;

deffunc H1( Element of X, Element of X) -> Element of X = o;

consider md being BinOp of X such that
Lemma13: for x, y being Element of X holds md . x,y = H1(x,y) from BINOP_1:sch 4();

E16: now
let F be Field;
set CV = [:X,X:];
defpred S1[ set ] means ex a, b being Element of X st
( a1 = [a,b] & b = o );
consider mo being set such that
E22: for x being set holds
( x in mo iff ( x in [:X,X:] & S1[x] ) ) from XBOOLE_0:sch 1();
mo c= [:X,X:]
proof
let x be set ; :: according to TARSKI:def 3
thus ( not x in mo or x in [:X,X:] ) by ;
end;
then reconsider mo = mo as Relation of X by RELSET_1:def 1;
take mo = mo;
thus for x being set holds
( x in mo iff ( x in [:X,X:] & ex a, b being Element of X st
( x = [a,b] & b = o ) ) ) by ;
end;

Lemma23: for F being Field
for mF being Function of [:the carrier of F,X:],X
for mo being Relation of X holds
( SymStr(# X,md,o,mF,mo #) is Abelian & SymStr(# X,md,o,mF,mo #) is add-associative & SymStr(# X,md,o,mF,mo #) is right_zeroed & SymStr(# X,md,o,mF,mo #) is right_complementable )
proof end;

E35: now
let F be Field;
let mF be Function of [:the carrier of F,X:],X;
assume E22: for a being Element of F
for x being Element of X holds mF . [a,x] = o ;
let mo be Relation of X;
let MPS be non empty Abelian add-associative right_zeroed right_complementable SymStr of F;
assume E26: MPS = SymStr(# X,md,o,mF,mo #) ;
thus MPS is VectSp-like
proof
for x, y being Element of F
for v, w being Element of MPS holds
( x * (v + w) = (x * v) + (x * w) & (x + y) * v = (x * v) + (y * v) & (x * y) * v = x * (y * v) & (1. F) * v = v )
proof
let x be Element of F, y be Element of F;
let v be Element of MPS, w be Element of MPS;
E27: x * (v + w) = (x * v) + (x * w)
proof
reconsider v = v, w = w as Element of MPS ;
E29: v + w = md . v,w by , RLVECT_1:5;
reconsider v = v, w = w as Element of X by ;
E30: md . v,w = o by ;
reconsider v = v, w = w as Element of MPS ;
E31: x * (v + w) = mF . x,o by , , , VECTSP_1:def 24;
E32: x * (v + w) = o by , ;
reconsider v = v as Element of MPS ;
E33: mF . x,v = o by ;
reconsider v = v as Element of MPS ;
E34: x * v = o by , , VECTSP_1:def 24;
reconsider w = w as Element of MPS ;
E39: mF . x,w = o by ;
reconsider w = w as Element of MPS ;
E40: x * w = o by , , VECTSP_1:def 24;
o = 0. MPS by , RLVECT_1:def 2;
hence x * (v + w) = (x * v) + (x * w) by , Lemma13, , RLVECT_1:10;
end;
E41: (x + y) * v = (x * v) + (y * v)
proof
set z = x + y;
E42: (x + y) * v = mF . (x + y),v by , VECTSP_1:def 24;
reconsider v = v as Element of MPS ;
reconsider v = v as Element of MPS ;
E43: (x + y) * v = o by , , ;
reconsider v = v as Element of MPS ;
E44: mF . x,v = o by , ;
reconsider v = v as Element of MPS ;
E45: x * v = o by , , VECTSP_1:def 24;
reconsider v = v as Element of MPS ;
E46: mF . y,v = o by , ;
reconsider v = v as Element of MPS ;
E47: y * v = o by , Lemma13, VECTSP_1:def 24;
o = 0. MPS by , RLVECT_1:def 2;
hence (x + y) * v = (x * v) + (y * v) by , , , RLVECT_1:10;
end;
E48: (x * y) * v = x * (y * v)
proof
set z = x * y;
E49: (x * y) * v = mF . (x * y),v by , VECTSP_1:def 24;
reconsider v = v as Element of MPS ;
reconsider v = v as Element of MPS ;
E50: (x * y) * v = o by , , ;
reconsider v = v as Element of MPS ;
E51: mF . y,v = o by , ;
reconsider v = v as Element of MPS ;
y * v = o by , , VECTSP_1:def 24;
then E52: x * (y * v) = mF . x,o by , VECTSP_1:def 24;
thus (x * y) * v = x * (y * v) by , , Lemma13;
end;
(1. F) * v = v
proof
set one = 1. F;
E54: (1. F) * v = mF . (1. F),v by , VECTSP_1:def 24;
reconsider v = v as Element of MPS ;
mF . (1. F),v = o by , ;
hence (1. F) * v = v by , , TARSKI:def 1;
end;
hence ( x * (v + w) = (x * v) + (x * w) & (x + y) * v = (x * v) + (y * v) & (x * y) * v = x * (y * v) & (1. F) * v = v ) by , , ;
end;
hence MPS is VectSp-like by VECTSP_1:def 26;
end;
end;

E55: now
let F be Field;
let mF be Function of [:the carrier of F,X:],X;
assume for a being Element of F
for x being Element of X holds mF . [a,x] = o ;
set CV = [:X,X:];
let mo be Relation of X;
assume E22: for x being set holds
( x in mo iff ( x in [:X,X:] & ex a, b being Element of X st
( x = [a,b] & b = o ) ) ) ;
let MPS be non empty Abelian add-associative right_zeroed right_complementable SymStr of F;
assume E26: MPS = SymStr(# X,md,o,mF,mo #) ;
E27: for a, b being Element of MPS holds
( a _|_ b iff ( [a,b] in [:X,X:] & ex c, d being Element of X st
( [a,b] = [c,d] & d = o ) ) )
proof
let a be Element of MPS, b be Element of MPS;
( a _|_ b iff [a,b] in mo ) by , SYMSP_1:def 1;
hence ( a _|_ b iff ( [a,b] in [:X,X:] & ex c, d being Element of X st
( [a,b] = [c,d] & d = o ) ) ) by ;
end;
E29: for a, b being Element of MPS holds
( a _|_ b iff b = o )
proof
let a be Element of MPS, b be Element of MPS;
E30: ( a _|_ b implies b = o )
proof
assume a _|_ b ;
then ex c, d being Element of X st
( [a,b] = [c,d] & d = o ) by ;
hence b = o by ZFMISC_1:33;
end;
( b = o implies a _|_ b )
proof
assume E31: b = o ;
consider c being Element of MPS, d being Element of MPS such that
E32: ( c = a & d = b ) ;
[a,b] = [c,d] by ;
hence a _|_ b by , , ;
end;
hence ( a _|_ b iff b = o ) by ;
end;
0. MPS = o by , TARSKI:def 1;
hence for a, b, c, d being Element of MPS st a <> 0. MPS & b <> 0. MPS & c <> 0. MPS & d <> 0. MPS holds
ex p being Element of MPS st
( not p _|_ a & not p _|_ b & not p _|_ c & not p _|_ d ) by , TARSKI:def 1;
thus for a, b being Element of MPS
for l being Element of F st a _|_ b holds
l * a _|_ b
proof
let a be Element of MPS, b be Element of MPS;
let l be Element of F;
assume a _|_ b ;
then b = o by ;
hence l * a _|_ b by ;
end;
thus for a, b, c being Element of MPS st b _|_ a & c _|_ a holds
b + c _|_ a
proof
let a be Element of MPS, b be Element of MPS, c be Element of MPS;
assume ( b _|_ a & c _|_ a ) ;
then a = o by ;
hence b + c _|_ a by ;
end;
thus for a, b, x being Element of MPS st not b _|_ a holds
ex k being Element of F st x - (k * b) _|_ a
proof
let a be Element of MPS, b be Element of MPS, x be Element of MPS;
assume E33: not b _|_ a ;
assume for k being Element of F holds not x - (k * b) _|_ a ;
a <> o by , ;
hence contradiction by , TARSKI:def 1;
end;
thus for a, b, c being Element of MPS st a _|_ b - c & b _|_ c - a holds
c _|_ a - b
proof
let a be Element of MPS, b be Element of MPS, c be Element of MPS;
assume ( a _|_ b - c & b _|_ c - a ) ;
assume not c _|_ a - b ;
then a - b <> o by ;
hence contradiction by , TARSKI:def 1;
end;
end;

definition
let F be Field;
let IT be non empty Abelian add-associative right_zeroed right_complementable SymStr of F;
canceled;
attr a2 is OrtSp-like means :Def2: :: ORTSP_1:def 2
for a, b, c, d, x being Element of o
for l being Element of X holds
( ( a <> 0. o & b <> 0. o & c <> 0. o & d <> 0. o implies ex p being Element of o st
( not p _|_ a & not p _|_ b & not p _|_ c & not p _|_ d ) ) & ( a _|_ b implies l * a _|_ b ) & ( b _|_ a & c _|_ a implies b + c _|_ a ) & ( not b _|_ a implies ex k being Element of X st x - (k * b) _|_ a ) & ( a _|_ b - c & b _|_ c - a implies c _|_ a - b ) );
end;

:: deftheorem Def1 ORTSP_1:def 1 :
canceled;

:: deftheorem Def2 defines OrtSp-like ORTSP_1:def 2 :
for F being Field
for IT being non empty Abelian add-associative right_zeroed right_complementable SymStr of F holds
( IT is OrtSp-like iff for a, b, c, d, x being Element of IT
for l being Element of F holds
( ( a <> 0. IT & b <> 0. IT & c <> 0. IT & d <> 0. IT implies ex p being Element of IT st
( not p _|_ a & not p _|_ b & not p _|_ c & not p _|_ d ) ) & ( a _|_ b implies l * a _|_ b ) & ( b _|_ a & c _|_ a implies b + c _|_ a ) & ( not b _|_ a implies ex k being Element of F st x - (k * b) _|_ a ) & ( a _|_ b - c & b _|_ c - a implies c _|_ a - b ) ) );

registration
let F be Field;
cluster non empty Abelian add-associative right_zeroed right_complementable VectSp-like strict OrtSp-like SymStr of a1;
existence
ex b1 being non empty Abelian add-associative right_zeroed right_complementable SymStr of F st
( b1 is OrtSp-like & b1 is VectSp-like & b1 is strict )
proof end;
end;

definition
let F be Field;
mode OrtSp of a1 is non empty Abelian add-associative right_zeroed right_complementable VectSp-like OrtSp-like SymStr of a1;
end;

theorem Th1: :: ORTSP_1:1
canceled;

theorem Th2: :: ORTSP_1:2
canceled;

theorem Th3: :: ORTSP_1:3
canceled;

theorem Th4: :: ORTSP_1:4
canceled;

theorem Th5: :: ORTSP_1:5
canceled;

theorem Th6: :: ORTSP_1:6
canceled;

theorem Th7: :: ORTSP_1:7
canceled;

theorem Th8: :: ORTSP_1:8
canceled;

theorem Th9: :: ORTSP_1:9
canceled;

theorem Th10: :: ORTSP_1:10
canceled;

theorem Th11: :: ORTSP_1:11
for F being Field
for S being OrtSp of F
for a being Element of S holds 0. S _|_ a
proof end;

theorem Th12: :: ORTSP_1:12
for F being Field
for S being OrtSp of F
for a, b being Element of S st a _|_ b holds
b _|_ a
proof end;

theorem Th13: :: ORTSP_1:13
for F being Field
for S being OrtSp of F
for a, b, c being Element of S st not a _|_ b & c + a _|_ b holds
not c _|_ b
proof end;

theorem Th14: :: ORTSP_1:14
for F being Field
for S being OrtSp of F
for b, a, c being Element of S st not b _|_ a & c _|_ a holds
not b + c _|_ a
proof end;

theorem Th15: :: ORTSP_1:15
for F being Field
for S being OrtSp of F
for b, a being Element of S
for l being Element of F st not b _|_ a & not l = 0. F holds
( not l * b _|_ a & not b _|_ l * a )
proof end;

theorem Th16: :: ORTSP_1:16
for F being Field
for S being OrtSp of F
for a, b being Element of S st a _|_ b holds
- a _|_ b
proof end;

theorem Th17: :: ORTSP_1:17
canceled;

theorem Th18: :: ORTSP_1:18
canceled;

theorem Th19: :: ORTSP_1:19
for F being Field
for S being OrtSp of F
for a, b, d, c being Element of S st a - b _|_ d & a - c _|_ d holds
b - c _|_ d
proof end;

theorem Th20: :: ORTSP_1:20
for F being Field
for S being OrtSp of F
for b, a, x being Element of S
for k, l being Element of F st not b _|_ a & x - (k * b) _|_ a & x - (l * b) _|_ a holds
k = l
proof end;

theorem Th21: :: ORTSP_1:21
for F being Field
for S being OrtSp of F
for a, b being Element of S st a _|_ a & b _|_ b holds
a + b _|_ a - b
proof end;

theorem Th22: :: ORTSP_1:22
for F being Field
for S being OrtSp of F st (1. F) + (1. F) <> 0. F & ex a being Element of S st a <> 0. S holds
ex b being Element of S st not b _|_ b
proof end;

definition
let F be Field;
let S be OrtSp of F;
let a be Element of S;
let b be Element of S;
let x be Element of S;
assume E22: not b _|_ a ;
canceled;
canceled;
canceled;
func ProJ c3,c4,c5 -> Element of a1 means :Def6: :: ORTSP_1:def 6
for l being Element of X st S - (l * F) _|_ md holds
it = l;
existence
ex b1 being Element of F st
for l being Element of F st x - (l * b) _|_ a holds
b1 = l
proof end;
uniqueness
for b1, b2 being Element of F st ( for l being Element of F st x - (l * b) _|_ a holds
b1 = l ) & ( for l being Element of F st x - (l * b) _|_ a holds
b2 = l ) holds
b1 = b2
proof end;
end;

:: deftheorem Def3 ORTSP_1:def 3 :
canceled;

:: deftheorem Def4 ORTSP_1:def 4 :
canceled;

:: deftheorem Def5 ORTSP_1:def 5 :
canceled;

:: deftheorem Def6 defines ProJ ORTSP_1:def 6 :
for F being Field
for S being OrtSp of F
for a, b, x being Element of S st not b _|_ a holds
for b6 being Element of F holds
( b6 = ProJ a,b,x iff for l being Element of F st x - (l * b) _|_ a holds
b6 = l );

theorem Th23: :: ORTSP_1:23
canceled;

theorem Th24: :: ORTSP_1:24
for F being Field
for S being OrtSp of F
for b, a, x being Element of S st not b _|_ a holds
x - ((ProJ a,b,x) * b) _|_ a
proof end;

theorem Th25: :: ORTSP_1:25
for F being Field
for S being OrtSp of F
for b, a, x being Element of S
for l being Element of F st not b _|_ a holds
ProJ a,b,(l * x) = l * (ProJ a,b,x)
proof end;

theorem Th26: :: ORTSP_1:26
for F being Field
for S being OrtSp of F
for b, a, x, y being Element of S st not b _|_ a holds
ProJ a,b,(x + y) = (ProJ a,b,x) + (ProJ a,b,y)
proof end;

theorem Th27: :: ORTSP_1:27
for F being Field
for S being OrtSp of F
for b, a, x being Element of S
for l being Element of F st not b _|_ a & l <> 0. F holds
ProJ a,(l * b),x = (l " ) * (ProJ a,b,x)
proof end;

theorem Th28: :: ORTSP_1:28
for F being Field
for S being OrtSp of F
for b, a, x being Element of S
for l being Element of F st not b _|_ a & l <> 0. F holds
ProJ (l * a),b,x = ProJ a,b,x
proof end;

theorem Th29: :: ORTSP_1:29
for F being Field
for S being OrtSp of F
for b, a, p, c being Element of S st not b _|_ a & p _|_ a holds
( ProJ a,(b + p),c = ProJ a,b,c & ProJ a,b,(c + p) = ProJ a,b,c )
proof end;

theorem Th30: :: ORTSP_1:30
for F being Field
for S being OrtSp of F
for b, a, p, c being Element of S st not b _|_ a & p _|_ b & p _|_ c holds
ProJ (a + p),b,c = ProJ a,b,c
proof end;

theorem Th31: :: ORTSP_1:31
for F being Field
for S being OrtSp of F
for b, a, c being Element of S st not b _|_ a & c - b _|_ a holds
ProJ a,b,c = 1. F
proof end;

theorem Th32: :: ORTSP_1:32
for F being Field
for S being OrtSp of F
for b, a being Element of S st not b _|_ a holds
ProJ a,b,b = 1. F
proof end;

theorem Th33: :: ORTSP_1:33
for F being Field
for S being OrtSp of F
for b, a, x being Element of S st not b _|_ a holds
( x _|_ a iff ProJ a,b,x = 0. F )
proof end;

theorem Th34: :: ORTSP_1:34
for F being Field
for S being OrtSp of F
for b, a, q, p being Element of S st not b _|_ a & not q _|_ a holds
(ProJ a,b,p) * ((ProJ a,b,q) " ) = ProJ a,q,p
proof end;

theorem Th35: :: ORTSP_1:35
for F being Field
for S being OrtSp of F
for b, a, c being Element of S st not b _|_ a & not c _|_ a holds
ProJ a,b,c = (ProJ a,c,b) "
proof end;

theorem Th36: :: ORTSP_1:36
for F being Field
for S being OrtSp of F
for b, a, c being Element of S st not b _|_ a & b _|_ c + a holds
ProJ a,b,c = - (ProJ c,b,a)
proof end;

theorem Th37: :: ORTSP_1:37
for F being Field
for S being OrtSp of F
for a, b, c being Element of S st not a _|_ b & not c _|_ b holds
ProJ c,b,a = ((ProJ b,a,c) " ) * (ProJ a,b,c)
proof end;

theorem Th38: :: ORTSP_1:38
for F being Field
for S being OrtSp of F
for p, a, x, q being Element of S st not p _|_ a & not p _|_ x & not q _|_ a & not q _|_ x holds
(ProJ a,q,p) * (ProJ p,a,x) = (ProJ q,a,x) * (ProJ x,q,p)
proof end;

theorem Th39: :: ORTSP_1:39
for F being Field
for S being OrtSp of F
for p, a, x, q, b, y being Element of S st not p _|_ a & not p _|_ x & not q _|_ a & not q _|_ x & not b _|_ a holds
((ProJ a,b,p) * (ProJ p,a,x)) * (ProJ x,p,y) = ((ProJ a,b,q) * (ProJ q,a,x)) * (ProJ x,q,y)
proof end;

theorem Th40: :: ORTSP_1:40
for F being Field
for S being OrtSp of F
for a, p, x, y being Element of S st not a _|_ p & not x _|_ p & not y _|_ p holds
(ProJ p,a,x) * (ProJ x,p,y) = (ProJ p,a,y) * (ProJ y,p,x)
proof end;

definition
let F be Field;
let S be OrtSp of F;
let x be Element of S;
let y be Element of S;
let a be Element of S;
let b be Element of S;
assume E22: not b _|_ a ;
func PProJ c5,c6,c3,c4 -> Element of a1 means :Def7: :: ORTSP_1:def 7
for q being Element of o st not q _|_ S & not q _|_ md holds
it = ((ProJ S,x,q) * (ProJ q,S,md)) * (ProJ md,q,F) if ex p being Element of o st
( not p _|_ S & not p _|_ md )
it = 0. X if for p being Element of o holds
( p _|_ S or p _|_ md )
;
existence
( ( ex p being Element of S st
( not p _|_ a & not p _|_ x ) implies ex b1 being Element of F st
for q being Element of S st not q _|_ a & not q _|_ x holds
b1 = ((ProJ a,b,q) * (ProJ q,a,x)) * (ProJ x,q,y) ) & ( ( for p being Element of S holds
( p _|_ a or p _|_ x ) ) implies ex b1 being Element of F st b1 = 0. F ) )
proof end;
uniqueness
for b1, b2 being Element of F holds
( ( ex p being Element of S st
( not p _|_ a & not p _|_ x ) & ( for q being Element of S st not q _|_ a & not q _|_ x holds
b1 = ((ProJ a,b,q) * (ProJ q,a,x)) * (ProJ x,q,y) ) & ( for q being Element of S st not q _|_ a & not q _|_ x holds
b2 = ((ProJ a,b,q) * (ProJ q,a,x)) * (ProJ x,q,y) ) implies b1 = b2 ) & ( ( for p being Element of S holds
( p _|_ a or p _|_ x ) ) & b1 = 0. F & b2 = 0. F implies b1 = b2 ) )
proof end;
consistency
for b1 being Element of F st ex p being Element of S st
( not p _|_ a & not p _|_ x ) & ( for p being Element of S holds
( p _|_ a or p _|_ x ) ) holds
( ( for q being Element of S st not q _|_ a & not q _|_ x holds
b1 = ((ProJ a,b,q) * (ProJ q,a,x)) * (ProJ x,q,y) ) iff b1 = 0. F )
;
end;

:: deftheorem Def7 defines PProJ ORTSP_1:def 7 :
for F being Field
for S being OrtSp of F
for x, y, a, b being Element of S st not b _|_ a holds
for b7 being Element of F holds
( ( ex p being Element of S st
( not p _|_ a & not p _|_ x ) implies ( b7 = PProJ a,b,x,y iff for q being Element of S st not q _|_ a & not q _|_ x holds
b7 = ((ProJ a,b,q) * (ProJ q,a,x)) * (ProJ x,q,y) ) ) & ( ( for p being Element of S holds
( p _|_ a or p _|_ x ) ) implies ( b7 = PProJ a,b,x,y iff b7 = 0. F ) ) );

theorem Th41: :: ORTSP_1:41
canceled;

theorem Th42: :: ORTSP_1:42
canceled;

theorem Th43: :: ORTSP_1:43
for F being Field
for S being OrtSp of F
for b, a, x, y being Element of S st not b _|_ a & x = 0. S holds
PProJ a,b,x,y = 0. F
proof end;

Lemma101: for F being Field
for S being OrtSp of F
for x being Element of S holds x _|_ 0. S
proof end;

theorem Th44: :: ORTSP_1:44
for F being Field
for S being OrtSp of F
for b, a, x, y being Element of S st not b _|_ a holds
( PProJ a,b,x,y = 0. F iff y _|_ x )
proof end;

theorem Th45: :: ORTSP_1:45
for F being Field
for S being OrtSp of F
for b, a, x, y being Element of S st not b _|_ a holds
PProJ a,b,x,y = PProJ a,b,y,x
proof end;

theorem Th46: :: ORTSP_1:46
for F being Field
for S being OrtSp of F
for b, a, x, y being Element of S
for l being Element of F st not b _|_ a holds
PProJ a,b,x,(l * y) = l * (PProJ a,b,x,y)
proof end;

theorem Th47: :: ORTSP_1:47
for F being Field
for S being OrtSp of F
for b, a, x, y, z being Element of S st not b _|_ a holds
PProJ a,b,x,(y + z) = (PProJ a,b,x,y) + (PProJ a,b,x,z)
proof end;