let X be set ; :: thesis: for O1, O2 being Operation of X st ( for x being Element of X holds x . O1 = x . O2 ) holds
O1 = O2
let O1, O2 be Operation of X; :: thesis: ( ( for x being Element of X holds x . O1 = x . O2 ) implies O1 = O2 )
assume Z0: for x being Element of X holds x . O1 = x . O2 ; :: thesis: O1 = O2
let a, b be set ; :: according to RELAT_1:def 2 :: thesis: ( ( not [a,b] in O1 or [a,b] in O2 ) & ( not [a,b] in O2 or [a,b] in O1 ) )
thus ( [a,b] in O1 implies [a,b] in O2 ) :: thesis: ( not [a,b] in O2 or [a,b] in O1 )

proof

assume A1: [a,b] in O1 ; :: thesis: [a,b] in O2
reconsider a = a, b = b as Element of X by A1, ZFMISC_1:87;
b in a . O1 by A1, RELAT_1:169;
then b in a . O2 by Z0;
hence [a,b] in O2 by RELAT_1:169; :: thesis: verum

end;

assume A1: [a,b] in O2 ; :: thesis: [a,b] in O1
then A2: ( a in X & b in X ) by ZFMISC_1:87;
reconsider a = a, b = b as Element of X by A1, ZFMISC_1:87;
reconsider L = {a} as Subset of X by A2, ZFMISC_1:31;
b in a . O2 by A1, RELAT_1:169;
then b in a . O1 by Z0;
hence [a,b] in O1 by RELAT_1:169; :: thesis: verum