set X = Morphs V;
defpred S₁[ Element of Morphs V, Element of Morphs V] means dom' $1 = cod' $2;
let c1, c2 be PartFunc of [:(Morphs V),(Morphs V):],(Morphs V); :: thesis: ( ( for g, f being Element of Morphs V holds
( [g,f] in dom c1 iff dom' g = cod' f ) ) & ( for g, f being Element of Morphs V st [g,f] in dom c1 holds
c1 . g,f = g * f ) & ( for g, f being Element of Morphs V holds
( [g,f] in dom c2 iff dom' g = cod' f ) ) & ( for g, f being Element of Morphs V st [g,f] in dom c2 holds
c2 . g,f = g * f ) implies c1 = c2 )
assume that
A4: for g, f being Element of Morphs V holds
( [g,f] in dom c1 iff S₁[g,f] ) and
A5: for g, f being Element of Morphs V st [g,f] in dom c1 holds
c1 . g,f = g * f and
A6: for g, f being Element of Morphs V holds
( [g,f] in dom c2 iff S₁[g,f] ) and
A7: for g, f being Element of Morphs V st [g,f] in dom c2 holds
c2 . g,f = g * f ; :: thesis: c1 = c2
A8: ( dom c1 c= [:(Morphs V),(Morphs V):] & dom c2 c= [:(Morphs V),(Morphs V):] ) by RELAT_1:def 18;
A9: dom c1 = dom c2

proof

now

let x be set ; :: thesis: ( x in dom c1 implies x in dom c2 )
assume A10: x in dom c1 ; :: thesis: x in dom c2
then consider g, f being Element of Morphs V such that
A11: x = [g,f] by A8, GRCAT_1:2;
S₁[g,f] by A4, A10, A11;
hence x in dom c2 by A6, A11; :: thesis: verum

end;

then A12: dom c1 c= dom c2 by TARSKI:def 3;

now

let x be set ; :: thesis: ( x in dom c2 implies x in dom c1 )
assume A13: x in dom c2 ; :: thesis: x in dom c1
then consider g, f being Element of Morphs V such that
A14: x = [g,f] by A8, GRCAT_1:2;
S₁[g,f] by A6, A13, A14;
hence x in dom c1 by A4, A14; :: thesis: verum

end;

then dom c2 c= dom c1 by TARSKI:def 3;
hence dom c1 = dom c2 by A12, XBOOLE_0:def 10; :: thesis: verum

end;

set V0 = dom c1;
for x, y being set st [x,y] in dom c1 holds
c1 . x,y = c2 . x,y

proof

let x, y be set ; :: thesis: ( [x,y] in dom c1 implies c1 . x,y = c2 . x,y )
assume A15: [x,y] in dom c1 ; :: thesis: c1 . x,y = c2 . x,y
then reconsider x = x, y = y as Element of Morphs V by A8, ZFMISC_1:106;
c1 . x,y = x * y by A5, A15;
hence c1 . x,y = c2 . x,y by A7, A9, A15; :: thesis: verum

end;

hence c1 = c2 by A9, BINOP_1:32; :: thesis: verum