let G1, G2 be RelStr ; :: thesis: ( the carrier of G1 misses the carrier of G2 implies ComplRelStr (sum_of (G1,G2)) = union_of ((ComplRelStr G1),(ComplRelStr G2)) )

assume A1: the carrier of G1 misses the carrier of G2 ; :: thesis: ComplRelStr (sum_of (G1,G2)) = union_of ((ComplRelStr G1),(ComplRelStr G2))

set P = the InternalRel of (ComplRelStr (sum_of (G1,G2)));

set R = the InternalRel of (union_of ((ComplRelStr G1),(ComplRelStr G2)));

set X1 = the InternalRel of (ComplRelStr G1);

set X2 = the InternalRel of (ComplRelStr G2);

set X5 = [: the carrier of G1, the carrier of G1:];

set X6 = [: the carrier of G2, the carrier of G2:];

set X7 = [: the carrier of G1, the carrier of G2:];

set X8 = [: the carrier of G2, the carrier of G1:];

A2: [: the carrier of (sum_of (G1,G2)), the carrier of (sum_of (G1,G2)):] = [:( the carrier of G1 \/ the carrier of G2), the carrier of (sum_of (G1,G2)):] by NECKLA_2:def 3

.= [:( the carrier of G1 \/ the carrier of G2),( the carrier of G1 \/ the carrier of G2):] by NECKLA_2:def 3

.= (([: the carrier of G1, the carrier of G1:] \/ [: the carrier of G1, the carrier of G2:]) \/ [: the carrier of G2, the carrier of G1:]) \/ [: the carrier of G2, the carrier of G2:] by ZFMISC_1:98 ;

A3: for a, b being object holds

( [a,b] in the InternalRel of (ComplRelStr (sum_of (G1,G2))) iff [a,b] in the InternalRel of (union_of ((ComplRelStr G1),(ComplRelStr G2))) )

.= the carrier of G1 \/ the carrier of (ComplRelStr G2) by NECKLACE:def 8

.= the carrier of G1 \/ the carrier of G2 by NECKLACE:def 8 ;

the carrier of (ComplRelStr (sum_of (G1,G2))) = the carrier of (sum_of (G1,G2)) by NECKLACE:def 8

.= the carrier of G1 \/ the carrier of G2 by NECKLA_2:def 3 ;

hence ComplRelStr (sum_of (G1,G2)) = union_of ((ComplRelStr G1),(ComplRelStr G2)) by A38, A3, RELAT_1:def 2; :: thesis: verum

assume A1: the carrier of G1 misses the carrier of G2 ; :: thesis: ComplRelStr (sum_of (G1,G2)) = union_of ((ComplRelStr G1),(ComplRelStr G2))

set P = the InternalRel of (ComplRelStr (sum_of (G1,G2)));

set R = the InternalRel of (union_of ((ComplRelStr G1),(ComplRelStr G2)));

set X1 = the InternalRel of (ComplRelStr G1);

set X2 = the InternalRel of (ComplRelStr G2);

set X5 = [: the carrier of G1, the carrier of G1:];

set X6 = [: the carrier of G2, the carrier of G2:];

set X7 = [: the carrier of G1, the carrier of G2:];

set X8 = [: the carrier of G2, the carrier of G1:];

A2: [: the carrier of (sum_of (G1,G2)), the carrier of (sum_of (G1,G2)):] = [:( the carrier of G1 \/ the carrier of G2), the carrier of (sum_of (G1,G2)):] by NECKLA_2:def 3

.= [:( the carrier of G1 \/ the carrier of G2),( the carrier of G1 \/ the carrier of G2):] by NECKLA_2:def 3

.= (([: the carrier of G1, the carrier of G1:] \/ [: the carrier of G1, the carrier of G2:]) \/ [: the carrier of G2, the carrier of G1:]) \/ [: the carrier of G2, the carrier of G2:] by ZFMISC_1:98 ;

A3: for a, b being object holds

( [a,b] in the InternalRel of (ComplRelStr (sum_of (G1,G2))) iff [a,b] in the InternalRel of (union_of ((ComplRelStr G1),(ComplRelStr G2))) )

proof

A38: the carrier of (union_of ((ComplRelStr G1),(ComplRelStr G2))) =
the carrier of (ComplRelStr G1) \/ the carrier of (ComplRelStr G2)
by NECKLA_2:def 2
let a, b be object ; :: thesis: ( [a,b] in the InternalRel of (ComplRelStr (sum_of (G1,G2))) iff [a,b] in the InternalRel of (union_of ((ComplRelStr G1),(ComplRelStr G2))) )

set x = [a,b];

thus ( [a,b] in the InternalRel of (ComplRelStr (sum_of (G1,G2))) implies [a,b] in the InternalRel of (union_of ((ComplRelStr G1),(ComplRelStr G2))) ) :: thesis: ( [a,b] in the InternalRel of (union_of ((ComplRelStr G1),(ComplRelStr G2))) implies [a,b] in the InternalRel of (ComplRelStr (sum_of (G1,G2))) )

then A11: [a,b] in the InternalRel of (ComplRelStr G1) \/ the InternalRel of (ComplRelStr G2) by NECKLA_2:def 2;

end;set x = [a,b];

thus ( [a,b] in the InternalRel of (ComplRelStr (sum_of (G1,G2))) implies [a,b] in the InternalRel of (union_of ((ComplRelStr G1),(ComplRelStr G2))) ) :: thesis: ( [a,b] in the InternalRel of (union_of ((ComplRelStr G1),(ComplRelStr G2))) implies [a,b] in the InternalRel of (ComplRelStr (sum_of (G1,G2))) )

proof

assume
[a,b] in the InternalRel of (union_of ((ComplRelStr G1),(ComplRelStr G2)))
; :: thesis: [a,b] in the InternalRel of (ComplRelStr (sum_of (G1,G2)))
assume
[a,b] in the InternalRel of (ComplRelStr (sum_of (G1,G2)))
; :: thesis: [a,b] in the InternalRel of (union_of ((ComplRelStr G1),(ComplRelStr G2)))

then A4: [a,b] in ( the InternalRel of (sum_of (G1,G2)) `) \ (id the carrier of (sum_of (G1,G2))) by NECKLACE:def 8;

then ( [a,b] in ([: the carrier of G1, the carrier of G1:] \/ [: the carrier of G1, the carrier of G2:]) \/ [: the carrier of G2, the carrier of G1:] or [a,b] in [: the carrier of G2, the carrier of G2:] ) by A2, XBOOLE_0:def 3;

then A5: ( [a,b] in [: the carrier of G1, the carrier of G1:] \/ [: the carrier of G1, the carrier of G2:] or [a,b] in [: the carrier of G2, the carrier of G1:] or [a,b] in [: the carrier of G2, the carrier of G2:] ) by XBOOLE_0:def 3;

[a,b] in the InternalRel of (sum_of (G1,G2)) ` by A4, XBOOLE_0:def 5;

then [a,b] in [: the carrier of (sum_of (G1,G2)), the carrier of (sum_of (G1,G2)):] \ the InternalRel of (sum_of (G1,G2)) by SUBSET_1:def 4;

then not [a,b] in the InternalRel of (sum_of (G1,G2)) by XBOOLE_0:def 5;

then A6: not [a,b] in (( the InternalRel of G1 \/ the InternalRel of G2) \/ [: the carrier of G1, the carrier of G2:]) \/ [: the carrier of G2, the carrier of G1:] by NECKLA_2:def 3;

A7: ( not [a,b] in the InternalRel of G1 & not [a,b] in the InternalRel of G2 & not [a,b] in [: the carrier of G1, the carrier of G2:] & not [a,b] in [: the carrier of G2, the carrier of G1:] )

then not [a,b] in id ( the carrier of G1 \/ the carrier of G2) by NECKLA_2:def 3;

then A8: not [a,b] in (id the carrier of G1) \/ (id the carrier of G2) by SYSREL:14;

then A9: not [a,b] in id the carrier of G1 by XBOOLE_0:def 3;

A10: not [a,b] in id the carrier of G2 by A8, XBOOLE_0:def 3;

end;then A4: [a,b] in ( the InternalRel of (sum_of (G1,G2)) `) \ (id the carrier of (sum_of (G1,G2))) by NECKLACE:def 8;

then ( [a,b] in ([: the carrier of G1, the carrier of G1:] \/ [: the carrier of G1, the carrier of G2:]) \/ [: the carrier of G2, the carrier of G1:] or [a,b] in [: the carrier of G2, the carrier of G2:] ) by A2, XBOOLE_0:def 3;

then A5: ( [a,b] in [: the carrier of G1, the carrier of G1:] \/ [: the carrier of G1, the carrier of G2:] or [a,b] in [: the carrier of G2, the carrier of G1:] or [a,b] in [: the carrier of G2, the carrier of G2:] ) by XBOOLE_0:def 3;

[a,b] in the InternalRel of (sum_of (G1,G2)) ` by A4, XBOOLE_0:def 5;

then [a,b] in [: the carrier of (sum_of (G1,G2)), the carrier of (sum_of (G1,G2)):] \ the InternalRel of (sum_of (G1,G2)) by SUBSET_1:def 4;

then not [a,b] in the InternalRel of (sum_of (G1,G2)) by XBOOLE_0:def 5;

then A6: not [a,b] in (( the InternalRel of G1 \/ the InternalRel of G2) \/ [: the carrier of G1, the carrier of G2:]) \/ [: the carrier of G2, the carrier of G1:] by NECKLA_2:def 3;

A7: ( not [a,b] in the InternalRel of G1 & not [a,b] in the InternalRel of G2 & not [a,b] in [: the carrier of G1, the carrier of G2:] & not [a,b] in [: the carrier of G2, the carrier of G1:] )

proof

not [a,b] in id the carrier of (sum_of (G1,G2))
by A4, XBOOLE_0:def 5;
assume
( [a,b] in the InternalRel of G1 or [a,b] in the InternalRel of G2 or [a,b] in [: the carrier of G1, the carrier of G2:] or [a,b] in [: the carrier of G2, the carrier of G1:] )
; :: thesis: contradiction

then ( [a,b] in the InternalRel of G1 \/ the InternalRel of G2 or [a,b] in [: the carrier of G1, the carrier of G2:] or [a,b] in [: the carrier of G2, the carrier of G1:] ) by XBOOLE_0:def 3;

then ( [a,b] in ( the InternalRel of G1 \/ the InternalRel of G2) \/ [: the carrier of G1, the carrier of G2:] or [a,b] in [: the carrier of G2, the carrier of G1:] ) by XBOOLE_0:def 3;

hence contradiction by A6, XBOOLE_0:def 3; :: thesis: verum

end;then ( [a,b] in the InternalRel of G1 \/ the InternalRel of G2 or [a,b] in [: the carrier of G1, the carrier of G2:] or [a,b] in [: the carrier of G2, the carrier of G1:] ) by XBOOLE_0:def 3;

then ( [a,b] in ( the InternalRel of G1 \/ the InternalRel of G2) \/ [: the carrier of G1, the carrier of G2:] or [a,b] in [: the carrier of G2, the carrier of G1:] ) by XBOOLE_0:def 3;

hence contradiction by A6, XBOOLE_0:def 3; :: thesis: verum

then not [a,b] in id ( the carrier of G1 \/ the carrier of G2) by NECKLA_2:def 3;

then A8: not [a,b] in (id the carrier of G1) \/ (id the carrier of G2) by SYSREL:14;

then A9: not [a,b] in id the carrier of G1 by XBOOLE_0:def 3;

A10: not [a,b] in id the carrier of G2 by A8, XBOOLE_0:def 3;

per cases
( [a,b] in [: the carrier of G1, the carrier of G1:] or [a,b] in [: the carrier of G1, the carrier of G2:] or [a,b] in [: the carrier of G2, the carrier of G1:] or [a,b] in [: the carrier of G2, the carrier of G2:] )
by A5, XBOOLE_0:def 3;

end;

suppose
[a,b] in [: the carrier of G1, the carrier of G1:]
; :: thesis: [a,b] in the InternalRel of (union_of ((ComplRelStr G1),(ComplRelStr G2)))

then
[a,b] in [: the carrier of G1, the carrier of G1:] \ the InternalRel of G1
by A7, XBOOLE_0:def 5;

then [a,b] in the InternalRel of G1 ` by SUBSET_1:def 4;

then [a,b] in ( the InternalRel of G1 `) \ (id the carrier of G1) by A9, XBOOLE_0:def 5;

then [a,b] in the InternalRel of (ComplRelStr G1) by NECKLACE:def 8;

then [a,b] in the InternalRel of (ComplRelStr G1) \/ the InternalRel of (ComplRelStr G2) by XBOOLE_0:def 3;

hence [a,b] in the InternalRel of (union_of ((ComplRelStr G1),(ComplRelStr G2))) by NECKLA_2:def 2; :: thesis: verum

end;then [a,b] in the InternalRel of G1 ` by SUBSET_1:def 4;

then [a,b] in ( the InternalRel of G1 `) \ (id the carrier of G1) by A9, XBOOLE_0:def 5;

then [a,b] in the InternalRel of (ComplRelStr G1) by NECKLACE:def 8;

then [a,b] in the InternalRel of (ComplRelStr G1) \/ the InternalRel of (ComplRelStr G2) by XBOOLE_0:def 3;

hence [a,b] in the InternalRel of (union_of ((ComplRelStr G1),(ComplRelStr G2))) by NECKLA_2:def 2; :: thesis: verum

suppose
[a,b] in [: the carrier of G1, the carrier of G2:]
; :: thesis: [a,b] in the InternalRel of (union_of ((ComplRelStr G1),(ComplRelStr G2)))

hence
[a,b] in the InternalRel of (union_of ((ComplRelStr G1),(ComplRelStr G2)))
by A7; :: thesis: verum

end;suppose
[a,b] in [: the carrier of G2, the carrier of G1:]
; :: thesis: [a,b] in the InternalRel of (union_of ((ComplRelStr G1),(ComplRelStr G2)))

hence
[a,b] in the InternalRel of (union_of ((ComplRelStr G1),(ComplRelStr G2)))
by A7; :: thesis: verum

end;suppose
[a,b] in [: the carrier of G2, the carrier of G2:]
; :: thesis: [a,b] in the InternalRel of (union_of ((ComplRelStr G1),(ComplRelStr G2)))

then
[a,b] in [: the carrier of G2, the carrier of G2:] \ the InternalRel of G2
by A7, XBOOLE_0:def 5;

then [a,b] in the InternalRel of G2 ` by SUBSET_1:def 4;

then [a,b] in ( the InternalRel of G2 `) \ (id the carrier of G2) by A10, XBOOLE_0:def 5;

then [a,b] in the InternalRel of (ComplRelStr G2) by NECKLACE:def 8;

then [a,b] in the InternalRel of (ComplRelStr G1) \/ the InternalRel of (ComplRelStr G2) by XBOOLE_0:def 3;

hence [a,b] in the InternalRel of (union_of ((ComplRelStr G1),(ComplRelStr G2))) by NECKLA_2:def 2; :: thesis: verum

end;then [a,b] in the InternalRel of G2 ` by SUBSET_1:def 4;

then [a,b] in ( the InternalRel of G2 `) \ (id the carrier of G2) by A10, XBOOLE_0:def 5;

then [a,b] in the InternalRel of (ComplRelStr G2) by NECKLACE:def 8;

then [a,b] in the InternalRel of (ComplRelStr G1) \/ the InternalRel of (ComplRelStr G2) by XBOOLE_0:def 3;

hence [a,b] in the InternalRel of (union_of ((ComplRelStr G1),(ComplRelStr G2))) by NECKLA_2:def 2; :: thesis: verum

then A11: [a,b] in the InternalRel of (ComplRelStr G1) \/ the InternalRel of (ComplRelStr G2) by NECKLA_2:def 2;

per cases
( [a,b] in the InternalRel of (ComplRelStr G1) or [a,b] in the InternalRel of (ComplRelStr G2) )
by A11, XBOOLE_0:def 3;

end;

suppose
[a,b] in the InternalRel of (ComplRelStr G1)
; :: thesis: [a,b] in the InternalRel of (ComplRelStr (sum_of (G1,G2)))

then A12:
[a,b] in ( the InternalRel of G1 `) \ (id the carrier of G1)
by NECKLACE:def 8;

then A13: not [a,b] in id the carrier of G1 by XBOOLE_0:def 5;

A14: not [a,b] in id the carrier of (sum_of (G1,G2))

then [a,b] in [: the carrier of G1, the carrier of G1:] \ the InternalRel of G1 by SUBSET_1:def 4;

then A16: not [a,b] in the InternalRel of G1 by XBOOLE_0:def 5;

A17: not [a,b] in the InternalRel of (sum_of (G1,G2))

then [a,b] in [: the carrier of (sum_of (G1,G2)), the carrier of (sum_of (G1,G2)):] by A2, XBOOLE_1:113;

then [a,b] in [: the carrier of (sum_of (G1,G2)), the carrier of (sum_of (G1,G2)):] \ the InternalRel of (sum_of (G1,G2)) by A17, XBOOLE_0:def 5;

then [a,b] in the InternalRel of (sum_of (G1,G2)) ` by SUBSET_1:def 4;

then [a,b] in ( the InternalRel of (sum_of (G1,G2)) `) \ (id the carrier of (sum_of (G1,G2))) by A14, XBOOLE_0:def 5;

hence [a,b] in the InternalRel of (ComplRelStr (sum_of (G1,G2))) by NECKLACE:def 8; :: thesis: verum

end;then A13: not [a,b] in id the carrier of G1 by XBOOLE_0:def 5;

A14: not [a,b] in id the carrier of (sum_of (G1,G2))

proof

[a,b] in the InternalRel of G1 `
by A12, XBOOLE_0:def 5;
assume
[a,b] in id the carrier of (sum_of (G1,G2))
; :: thesis: contradiction

then [a,b] in id ( the carrier of G1 \/ the carrier of G2) by NECKLA_2:def 3;

then [a,b] in (id the carrier of G1) \/ (id the carrier of G2) by SYSREL:14;

then [a,b] in id the carrier of G2 by A13, XBOOLE_0:def 3;

then A15: a in the carrier of G2 by ZFMISC_1:87;

a in the carrier of G1 by A12, ZFMISC_1:87;

then not the carrier of G1 /\ the carrier of G2 is empty by A15, XBOOLE_0:def 4;

hence contradiction by A1; :: thesis: verum

end;then [a,b] in id ( the carrier of G1 \/ the carrier of G2) by NECKLA_2:def 3;

then [a,b] in (id the carrier of G1) \/ (id the carrier of G2) by SYSREL:14;

then [a,b] in id the carrier of G2 by A13, XBOOLE_0:def 3;

then A15: a in the carrier of G2 by ZFMISC_1:87;

a in the carrier of G1 by A12, ZFMISC_1:87;

then not the carrier of G1 /\ the carrier of G2 is empty by A15, XBOOLE_0:def 4;

hence contradiction by A1; :: thesis: verum

then [a,b] in [: the carrier of G1, the carrier of G1:] \ the InternalRel of G1 by SUBSET_1:def 4;

then A16: not [a,b] in the InternalRel of G1 by XBOOLE_0:def 5;

A17: not [a,b] in the InternalRel of (sum_of (G1,G2))

proof

[a,b] in [: the carrier of G1, the carrier of G1:] \/ (([: the carrier of G1, the carrier of G2:] \/ [: the carrier of G2, the carrier of G1:]) \/ [: the carrier of G2, the carrier of G2:])
by A12, XBOOLE_0:def 3;
assume
[a,b] in the InternalRel of (sum_of (G1,G2))
; :: thesis: contradiction

then [a,b] in (( the InternalRel of G1 \/ the InternalRel of G2) \/ [: the carrier of G1, the carrier of G2:]) \/ [: the carrier of G2, the carrier of G1:] by NECKLA_2:def 3;

then ( [a,b] in ( the InternalRel of G1 \/ the InternalRel of G2) \/ [: the carrier of G1, the carrier of G2:] or [a,b] in [: the carrier of G2, the carrier of G1:] ) by XBOOLE_0:def 3;

then A18: ( [a,b] in the InternalRel of G1 \/ the InternalRel of G2 or [a,b] in [: the carrier of G1, the carrier of G2:] or [a,b] in [: the carrier of G2, the carrier of G1:] ) by XBOOLE_0:def 3;

end;then [a,b] in (( the InternalRel of G1 \/ the InternalRel of G2) \/ [: the carrier of G1, the carrier of G2:]) \/ [: the carrier of G2, the carrier of G1:] by NECKLA_2:def 3;

then ( [a,b] in ( the InternalRel of G1 \/ the InternalRel of G2) \/ [: the carrier of G1, the carrier of G2:] or [a,b] in [: the carrier of G2, the carrier of G1:] ) by XBOOLE_0:def 3;

then A18: ( [a,b] in the InternalRel of G1 \/ the InternalRel of G2 or [a,b] in [: the carrier of G1, the carrier of G2:] or [a,b] in [: the carrier of G2, the carrier of G1:] ) by XBOOLE_0:def 3;

per cases
( [a,b] in the InternalRel of G2 or [a,b] in [: the carrier of G1, the carrier of G2:] or [a,b] in [: the carrier of G2, the carrier of G1:] )
by A16, A18, XBOOLE_0:def 3;

end;

suppose A19:
[a,b] in the InternalRel of G2
; :: thesis: contradiction

A20:
a in the carrier of G1
by A12, ZFMISC_1:87;

a in the carrier of G2 by A19, ZFMISC_1:87;

then not the carrier of G1 /\ the carrier of G2 is empty by A20, XBOOLE_0:def 4;

hence contradiction by A1; :: thesis: verum

end;a in the carrier of G2 by A19, ZFMISC_1:87;

then not the carrier of G1 /\ the carrier of G2 is empty by A20, XBOOLE_0:def 4;

hence contradiction by A1; :: thesis: verum

suppose A21:
[a,b] in [: the carrier of G1, the carrier of G2:]
; :: thesis: contradiction

A22:
b in the carrier of G1
by A12, ZFMISC_1:87;

b in the carrier of G2 by A21, ZFMISC_1:87;

then not the carrier of G1 /\ the carrier of G2 is empty by A22, XBOOLE_0:def 4;

hence contradiction by A1; :: thesis: verum

end;b in the carrier of G2 by A21, ZFMISC_1:87;

then not the carrier of G1 /\ the carrier of G2 is empty by A22, XBOOLE_0:def 4;

hence contradiction by A1; :: thesis: verum

suppose A23:
[a,b] in [: the carrier of G2, the carrier of G1:]
; :: thesis: contradiction

A24:
a in the carrier of G1
by A12, ZFMISC_1:87;

a in the carrier of G2 by A23, ZFMISC_1:87;

then not the carrier of G1 /\ the carrier of G2 is empty by A24, XBOOLE_0:def 4;

hence contradiction by A1; :: thesis: verum

end;a in the carrier of G2 by A23, ZFMISC_1:87;

then not the carrier of G1 /\ the carrier of G2 is empty by A24, XBOOLE_0:def 4;

hence contradiction by A1; :: thesis: verum

then [a,b] in [: the carrier of (sum_of (G1,G2)), the carrier of (sum_of (G1,G2)):] by A2, XBOOLE_1:113;

then [a,b] in [: the carrier of (sum_of (G1,G2)), the carrier of (sum_of (G1,G2)):] \ the InternalRel of (sum_of (G1,G2)) by A17, XBOOLE_0:def 5;

then [a,b] in the InternalRel of (sum_of (G1,G2)) ` by SUBSET_1:def 4;

then [a,b] in ( the InternalRel of (sum_of (G1,G2)) `) \ (id the carrier of (sum_of (G1,G2))) by A14, XBOOLE_0:def 5;

hence [a,b] in the InternalRel of (ComplRelStr (sum_of (G1,G2))) by NECKLACE:def 8; :: thesis: verum

suppose
[a,b] in the InternalRel of (ComplRelStr G2)
; :: thesis: [a,b] in the InternalRel of (ComplRelStr (sum_of (G1,G2)))

then A25:
[a,b] in ( the InternalRel of G2 `) \ (id the carrier of G2)
by NECKLACE:def 8;

then A26: not [a,b] in id the carrier of G2 by XBOOLE_0:def 5;

A27: not [a,b] in id the carrier of (sum_of (G1,G2))

then [a,b] in [: the carrier of G2, the carrier of G2:] \ the InternalRel of G2 by SUBSET_1:def 4;

then A29: not [a,b] in the InternalRel of G2 by XBOOLE_0:def 5;

A30: not [a,b] in the InternalRel of (sum_of (G1,G2))

then [a,b] in [: the carrier of (sum_of (G1,G2)), the carrier of (sum_of (G1,G2)):] \ the InternalRel of (sum_of (G1,G2)) by A30, XBOOLE_0:def 5;

then [a,b] in the InternalRel of (sum_of (G1,G2)) ` by SUBSET_1:def 4;

then [a,b] in ( the InternalRel of (sum_of (G1,G2)) `) \ (id the carrier of (sum_of (G1,G2))) by A27, XBOOLE_0:def 5;

hence [a,b] in the InternalRel of (ComplRelStr (sum_of (G1,G2))) by NECKLACE:def 8; :: thesis: verum

end;then A26: not [a,b] in id the carrier of G2 by XBOOLE_0:def 5;

A27: not [a,b] in id the carrier of (sum_of (G1,G2))

proof

[a,b] in the InternalRel of G2 `
by A25, XBOOLE_0:def 5;
assume
[a,b] in id the carrier of (sum_of (G1,G2))
; :: thesis: contradiction

then [a,b] in id ( the carrier of G1 \/ the carrier of G2) by NECKLA_2:def 3;

then [a,b] in (id the carrier of G1) \/ (id the carrier of G2) by SYSREL:14;

then [a,b] in id the carrier of G1 by A26, XBOOLE_0:def 3;

then A28: b in the carrier of G1 by ZFMISC_1:87;

b in the carrier of G2 by A25, ZFMISC_1:87;

then not the carrier of G1 /\ the carrier of G2 is empty by A28, XBOOLE_0:def 4;

hence contradiction by A1; :: thesis: verum

end;then [a,b] in id ( the carrier of G1 \/ the carrier of G2) by NECKLA_2:def 3;

then [a,b] in (id the carrier of G1) \/ (id the carrier of G2) by SYSREL:14;

then [a,b] in id the carrier of G1 by A26, XBOOLE_0:def 3;

then A28: b in the carrier of G1 by ZFMISC_1:87;

b in the carrier of G2 by A25, ZFMISC_1:87;

then not the carrier of G1 /\ the carrier of G2 is empty by A28, XBOOLE_0:def 4;

hence contradiction by A1; :: thesis: verum

then [a,b] in [: the carrier of G2, the carrier of G2:] \ the InternalRel of G2 by SUBSET_1:def 4;

then A29: not [a,b] in the InternalRel of G2 by XBOOLE_0:def 5;

A30: not [a,b] in the InternalRel of (sum_of (G1,G2))

proof

[a,b] in [: the carrier of (sum_of (G1,G2)), the carrier of (sum_of (G1,G2)):]
by A2, A25, XBOOLE_0:def 3;
assume
[a,b] in the InternalRel of (sum_of (G1,G2))
; :: thesis: contradiction

then [a,b] in (( the InternalRel of G1 \/ the InternalRel of G2) \/ [: the carrier of G1, the carrier of G2:]) \/ [: the carrier of G2, the carrier of G1:] by NECKLA_2:def 3;

then ( [a,b] in ( the InternalRel of G1 \/ the InternalRel of G2) \/ [: the carrier of G1, the carrier of G2:] or [a,b] in [: the carrier of G2, the carrier of G1:] ) by XBOOLE_0:def 3;

then A31: ( [a,b] in the InternalRel of G1 \/ the InternalRel of G2 or [a,b] in [: the carrier of G1, the carrier of G2:] or [a,b] in [: the carrier of G2, the carrier of G1:] ) by XBOOLE_0:def 3;

end;then [a,b] in (( the InternalRel of G1 \/ the InternalRel of G2) \/ [: the carrier of G1, the carrier of G2:]) \/ [: the carrier of G2, the carrier of G1:] by NECKLA_2:def 3;

then ( [a,b] in ( the InternalRel of G1 \/ the InternalRel of G2) \/ [: the carrier of G1, the carrier of G2:] or [a,b] in [: the carrier of G2, the carrier of G1:] ) by XBOOLE_0:def 3;

then A31: ( [a,b] in the InternalRel of G1 \/ the InternalRel of G2 or [a,b] in [: the carrier of G1, the carrier of G2:] or [a,b] in [: the carrier of G2, the carrier of G1:] ) by XBOOLE_0:def 3;

per cases
( [a,b] in the InternalRel of G1 or [a,b] in [: the carrier of G1, the carrier of G2:] or [a,b] in [: the carrier of G2, the carrier of G1:] )
by A29, A31, XBOOLE_0:def 3;

end;

suppose A32:
[a,b] in the InternalRel of G1
; :: thesis: contradiction

A33:
a in the carrier of G2
by A25, ZFMISC_1:87;

a in the carrier of G1 by A32, ZFMISC_1:87;

then not the carrier of G1 /\ the carrier of G2 is empty by A33, XBOOLE_0:def 4;

hence contradiction by A1; :: thesis: verum

end;a in the carrier of G1 by A32, ZFMISC_1:87;

then not the carrier of G1 /\ the carrier of G2 is empty by A33, XBOOLE_0:def 4;

hence contradiction by A1; :: thesis: verum

suppose A34:
[a,b] in [: the carrier of G1, the carrier of G2:]
; :: thesis: contradiction

A35:
a in the carrier of G2
by A25, ZFMISC_1:87;

a in the carrier of G1 by A34, ZFMISC_1:87;

then not the carrier of G1 /\ the carrier of G2 is empty by A35, XBOOLE_0:def 4;

hence contradiction by A1; :: thesis: verum

end;a in the carrier of G1 by A34, ZFMISC_1:87;

then not the carrier of G1 /\ the carrier of G2 is empty by A35, XBOOLE_0:def 4;

hence contradiction by A1; :: thesis: verum

suppose A36:
[a,b] in [: the carrier of G2, the carrier of G1:]
; :: thesis: contradiction

A37:
b in the carrier of G2
by A25, ZFMISC_1:87;

b in the carrier of G1 by A36, ZFMISC_1:87;

then not the carrier of G1 /\ the carrier of G2 is empty by A37, XBOOLE_0:def 4;

hence contradiction by A1; :: thesis: verum

end;b in the carrier of G1 by A36, ZFMISC_1:87;

then not the carrier of G1 /\ the carrier of G2 is empty by A37, XBOOLE_0:def 4;

hence contradiction by A1; :: thesis: verum

then [a,b] in [: the carrier of (sum_of (G1,G2)), the carrier of (sum_of (G1,G2)):] \ the InternalRel of (sum_of (G1,G2)) by A30, XBOOLE_0:def 5;

then [a,b] in the InternalRel of (sum_of (G1,G2)) ` by SUBSET_1:def 4;

then [a,b] in ( the InternalRel of (sum_of (G1,G2)) `) \ (id the carrier of (sum_of (G1,G2))) by A27, XBOOLE_0:def 5;

hence [a,b] in the InternalRel of (ComplRelStr (sum_of (G1,G2))) by NECKLACE:def 8; :: thesis: verum

.= the carrier of G1 \/ the carrier of (ComplRelStr G2) by NECKLACE:def 8

.= the carrier of G1 \/ the carrier of G2 by NECKLACE:def 8 ;

the carrier of (ComplRelStr (sum_of (G1,G2))) = the carrier of (sum_of (G1,G2)) by NECKLACE:def 8

.= the carrier of G1 \/ the carrier of G2 by NECKLA_2:def 3 ;

hence ComplRelStr (sum_of (G1,G2)) = union_of ((ComplRelStr G1),(ComplRelStr G2)) by A38, A3, RELAT_1:def 2; :: thesis: verum