:: GOBOARD3 semantic presentation

E36: now
let f be FinSequence of (TOP-REAL 2);
let k be Element of NAT ;
assume E37: len f = k + 1 ;
assume k <> 0 ;
then E38: ( 0 < k & k <= k + 1 ) by NAT_1:29;
then ( 0 + 1 <= k & k <= len f & k + 1 <= len f ) by , NAT_1:38;
then E39: k in dom f by FINSEQ_3:27;
E40: len (f | k) = k by , , FINSEQ_1:80;
E41: dom (f | k) = Seg (len (f | k)) by FINSEQ_1:def 3;
assume E42: f is unfolded ;
thus f | k is unfolded
proof
set f1 = f | k;
let n be Element of NAT ; :: according to TOPREAL1:def 8
assume E44: ( 1 <= n & n + 2 <= len (f | k) ) ;
then ( n in dom (f | k) & n + 1 in dom (f | k) & n + 2 in dom (f | k) & (n + 1) + 1 = n + (1 + 1) ) by GOBOARD2:4;
then E45: ( LSeg (f | k),n = LSeg f,n & LSeg (f | k),(n + 1) = LSeg f,(n + 1) & (f | k) /. (n + 1) = f /. (n + 1) ) by , , , FINSEQ_4:86, TOPREAL3:24;
len (f | k) <= len f by , , FINSEQ_1:80;
then n + 2 <= len f by , XXREAL_0:2;
hence (LSeg (f | k),n) /\ (LSeg (f | k),(n + 1)) = {((f | k) /. (n + 1))} by , , , TOPREAL1:def 8;
end;
end;

theorem Th1: :: GOBOARD3:1
for f being FinSequence of (TOP-REAL 2)
for G being Go-board st ( for n being Element of NAT st n in dom f holds
ex i, j being Element of NAT st
( [i,j] in Indices G & f /. n = G * i,j ) ) & f is one-to-one & f is unfolded & f is s.n.c. & f is special holds
ex g being FinSequence of (TOP-REAL 2) st
( g is_sequence_on G & g is one-to-one & g is unfolded & g is s.n.c. & g is special & L~ f = L~ g & f /. 1 = g /. 1 & f /. (len f) = g /. (len g) & len f <= len g )
proof end;

theorem Th2: :: GOBOARD3:2
for f being FinSequence of (TOP-REAL 2)
for G being Go-board st ( for n being Element of NAT st n in dom f holds
ex i, j being Element of NAT st
( [i,j] in Indices G & f /. n = G * i,j ) ) & f is_S-Seq holds
ex g being FinSequence of (TOP-REAL 2) st
( g is_sequence_on G & g is_S-Seq & L~ f = L~ g & f /. 1 = g /. 1 & f /. (len f) = g /. (len g) & len f <= len g )
proof end;