let G1, G2 be Graph; :: thesis: ( the Source of G1 tolerates the Source of G2 & the Target of G1 tolerates the Target of G2 implies G1 \/ G2 = G2 \/ G1 )
assume A1: ( the Source of G1 tolerates the Source of G2 & the Target of G1 tolerates the Target of G2 ) ; :: thesis: G1 \/ G2 = G2 \/ G1
then A2: the carrier of (G1 \/ G2) = the carrier of G2 \/ the carrier of G1 by Def3
.= the carrier of (G2 \/ G1) by A1, Def3 ;
A3: the carrier' of (G1 \/ G2) = the carrier' of G2 \/ the carrier' of G1 by A1, Def3
.= the carrier' of (G2 \/ G1) by A1, Def3 ;
then A4: dom the Source of (G1 \/ G2) = the carrier' of (G2 \/ G1) by FUNCT_2:def 1
.= dom the Source of (G2 \/ G1) by FUNCT_2:def 1 ;
for v being set st v in dom the Source of (G1 \/ G2) holds
the Source of (G1 \/ G2) . v = the Source of (G2 \/ G1) . v

proof

let v be set ; :: thesis: ( v in dom the Source of (G1 \/ G2) implies the Source of (G1 \/ G2) . v = the Source of (G2 \/ G1) . v )
assume v in dom the Source of (G1 \/ G2) ; :: thesis: the Source of (G1 \/ G2) . v = the Source of (G2 \/ G1) . v
then v in the carrier' of (G1 \/ G2) ;
then A5: v in the carrier' of G1 \/ the carrier' of G2 by A1, Def3;

A6: now

assume A7: v in the carrier' of G1 ; :: thesis: the Source of (G1 \/ G2) . v = the Source of (G2 \/ G1) . v
hence the Source of (G1 \/ G2) . v = the Source of G1 . v by A1, Def3
.= the Source of (G2 \/ G1) . v by A1, A7, Def3 ;
:: thesis: verum

end;

now

assume A8: v in the carrier' of G2 ; :: thesis: the Source of (G1 \/ G2) . v = the Source of (G2 \/ G1) . v
hence the Source of (G1 \/ G2) . v = the Source of G2 . v by A1, Def3
.= the Source of (G2 \/ G1) . v by A1, A8, Def3 ;
:: thesis: verum

end;

hence the Source of (G1 \/ G2) . v = the Source of (G2 \/ G1) . v by A5, A6, XBOOLE_0:def 3; :: thesis: verum

end;

then A9: the Source of (G1 \/ G2) = the Source of (G2 \/ G1) by A4, FUNCT_1:2;
A10: dom the Target of (G1 \/ G2) = the carrier' of (G2 \/ G1) by A3, FUNCT_2:def 1
.= dom the Target of (G2 \/ G1) by FUNCT_2:def 1 ;
for v being set st v in dom the Target of (G1 \/ G2) holds
the Target of (G1 \/ G2) . v = the Target of (G2 \/ G1) . v

proof

let v be set ; :: thesis: ( v in dom the Target of (G1 \/ G2) implies the Target of (G1 \/ G2) . v = the Target of (G2 \/ G1) . v )
assume v in dom the Target of (G1 \/ G2) ; :: thesis: the Target of (G1 \/ G2) . v = the Target of (G2 \/ G1) . v
then v in the carrier' of (G1 \/ G2) ;
then A11: v in the carrier' of G1 \/ the carrier' of G2 by A1, Def3;

A12: now

assume A13: v in the carrier' of G1 ; :: thesis: the Target of (G1 \/ G2) . v = the Target of (G2 \/ G1) . v
hence the Target of (G1 \/ G2) . v = the Target of G1 . v by A1, Def3
.= the Target of (G2 \/ G1) . v by A1, A13, Def3 ;
:: thesis: verum

end;

now

assume A14: v in the carrier' of G2 ; :: thesis: the Target of (G1 \/ G2) . v = the Target of (G2 \/ G1) . v
hence the Target of (G1 \/ G2) . v = the Target of G2 . v by A1, Def3
.= the Target of (G2 \/ G1) . v by A1, A14, Def3 ;
:: thesis: verum

end;

hence the Target of (G1 \/ G2) . v = the Target of (G2 \/ G1) . v by A11, A12, XBOOLE_0:def 3; :: thesis: verum

end;

hence G1 \/ G2 = G2 \/ G1 by A2, A3, A9, A10, FUNCT_1:2; :: thesis: verum