:: BVFUNC11 semantic presentation

theorem Th1: :: BVFUNC11:1
for Y being non empty set
for z being Element of Y
for PA, PB being a_partition of Y st PA '<' PB holds
EqClass z,PA c= EqClass z,PB
proof end;

theorem :: BVFUNC11:2
for Y being non empty set
for z being Element of Y
for PA, PB being a_partition of Y holds EqClass z,PA c= EqClass z,(PA '\/' PB)
proof end;

theorem :: BVFUNC11:3
for Y being non empty set
for z being Element of Y
for PA, PB being a_partition of Y holds EqClass z,(PA '/\' PB) c= EqClass z,PA
proof end;

theorem :: BVFUNC11:4
for Y being non empty set
for z being Element of Y
for PA being a_partition of Y holds
( EqClass z,PA c= EqClass z,(%O Y) & EqClass z,(%I Y) c= EqClass z,PA )
proof end;

theorem :: BVFUNC11:5
for Y being non empty set
for G being Subset of (PARTITIONS Y)
for A, B being a_partition of Y st G is independent & G = {A,B} & A <> B holds
for a, b being set st a in A & b in B holds
a meets b
proof end;

theorem :: BVFUNC11:6
for Y being non empty set
for G being Subset of (PARTITIONS Y)
for A, B being a_partition of Y st G is independent & G = {A,B} & A <> B holds
'/\' G = A '/\' B
proof end;

theorem :: BVFUNC11:7
for Y being non empty set
for G being Subset of (PARTITIONS Y)
for A, B being a_partition of Y st G = {A,B} & A <> B holds
CompF A,G = B
proof end;

theorem Th8: :: BVFUNC11:8
for Y being non empty set
for a, b being Element of Funcs Y,BOOLEAN
for G being Subset of (PARTITIONS Y)
for A being a_partition of Y st a '<' b holds
All a,A,G '<' Ex b,A,G
proof end;

theorem :: BVFUNC11:9
canceled;

theorem :: BVFUNC11:10
canceled;

theorem :: BVFUNC11:11
for Y being non empty set
for a being Element of Funcs Y,BOOLEAN
for G being Subset of (PARTITIONS Y)
for A, B being a_partition of Y st G is independent holds
All (All a,A,G),B,G '<' Ex (All a,B,G),A,G
proof end;

theorem :: BVFUNC11:12
for Y being non empty set
for a being Element of Funcs Y,BOOLEAN
for G being Subset of (PARTITIONS Y)
for A, B being a_partition of Y holds All (All a,A,G),B,G '<' Ex (Ex a,B,G),A,G
proof end;

theorem :: BVFUNC11:13
for Y being non empty set
for a being Element of Funcs Y,BOOLEAN
for G being Subset of (PARTITIONS Y)
for A, B being a_partition of Y st G is independent holds
All (All a,A,G),B,G '<' All (Ex a,B,G),A,G
proof end;

theorem :: BVFUNC11:14
for Y being non empty set
for a being Element of Funcs Y,BOOLEAN
for G being Subset of (PARTITIONS Y)
for A, B being a_partition of Y holds All (Ex a,A,G),B,G '<' Ex (Ex a,B,G),A,G
proof end;

theorem :: BVFUNC11:15
for Y being non empty set
for a being Element of Funcs Y,BOOLEAN
for G being Subset of (PARTITIONS Y)
for A, B being a_partition of Y holds 'not' (Ex (All a,A,G),B,G) '<' Ex (Ex ('not' a),B,G),A,G
proof end;

theorem Th16: :: BVFUNC11:16
for Y being non empty set
for a being Element of Funcs Y,BOOLEAN
for G being Subset of (PARTITIONS Y)
for A, B being a_partition of Y st G is independent holds
Ex ('not' (All a,A,G)),B,G '<' Ex (Ex ('not' a),B,G),A,G
proof end;

theorem Th17: :: BVFUNC11:17
for Y being non empty set
for a being Element of Funcs Y,BOOLEAN
for G being Subset of (PARTITIONS Y)
for A, B being a_partition of Y st G is independent holds
'not' (All (All a,A,G),B,G) = Ex ('not' (All a,B,G)),A,G
proof end;

theorem :: BVFUNC11:18
for Y being non empty set
for a being Element of Funcs Y,BOOLEAN
for G being Subset of (PARTITIONS Y)
for A, B being a_partition of Y st G is independent holds
All ('not' (All a,A,G)),B,G '<' Ex (Ex ('not' a),B,G),A,G
proof end;

theorem :: BVFUNC11:19
for Y being non empty set
for a being Element of Funcs Y,BOOLEAN
for G being Subset of (PARTITIONS Y)
for A, B being a_partition of Y st G is independent holds
'not' (All (All a,A,G),B,G) = Ex (Ex ('not' a),B,G),A,G
proof end;

theorem :: BVFUNC11:20
for Y being non empty set
for a being Element of Funcs Y,BOOLEAN
for G being Subset of (PARTITIONS Y)
for A, B being a_partition of Y st G is independent holds
'not' (All (All a,A,G),B,G) '<' Ex (Ex ('not' a),A,G),B,G
proof end;

theorem :: BVFUNC11:21
for Y being non empty set
for a being Element of Funcs Y,BOOLEAN
for G being Subset of (PARTITIONS Y)
for A, B being a_partition of Y holds 'not' (All (Ex a,A,G),B,G) = Ex (All ('not' a),A,G),B,G
proof end;

theorem :: BVFUNC11:22
for Y being non empty set
for a being Element of Funcs Y,BOOLEAN
for G being Subset of (PARTITIONS Y)
for A, B being a_partition of Y holds 'not' (Ex (All a,A,G),B,G) = All (Ex ('not' a),A,G),B,G
proof end;

theorem :: BVFUNC11:23
for Y being non empty set
for a being Element of Funcs Y,BOOLEAN
for G being Subset of (PARTITIONS Y)
for A, B being a_partition of Y holds 'not' (All (All a,A,G),B,G) = Ex (Ex ('not' a),A,G),B,G
proof end;

theorem :: BVFUNC11:24
for Y being non empty set
for a being Element of Funcs Y,BOOLEAN
for G being Subset of (PARTITIONS Y)
for A, B being a_partition of Y st G is independent holds
Ex (All a,A,G),B,G '<' Ex (Ex a,B,G),A,G
proof end;

theorem :: BVFUNC11:25
for Y being non empty set
for a being Element of Funcs Y,BOOLEAN
for G being Subset of (PARTITIONS Y)
for A, B being a_partition of Y holds All (All a,A,G),B,G '<' All (Ex a,A,G),B,G
proof end;

theorem :: BVFUNC11:26
for Y being non empty set
for a being Element of Funcs Y,BOOLEAN
for G being Subset of (PARTITIONS Y)
for A, B being a_partition of Y holds All (All a,A,G),B,G '<' Ex (Ex a,A,G),B,G
proof end;

theorem :: BVFUNC11:27
for Y being non empty set
for a being Element of Funcs Y,BOOLEAN
for G being Subset of (PARTITIONS Y)
for A, B being a_partition of Y holds Ex (All a,A,G),B,G '<' Ex (Ex a,A,G),B,G
proof end;