reserve X,Y,z,s for set, L,L1,L2,A,B for List of X, x for Element of X,
  O,O1,O2,O3 for Operation of X, a,b,y for Element of X, n,m for Nat;

theorem Th29:
  for O1,O2 being Operation of X st for x holds x.O1 = x.O2 holds O1 = O2
  proof
    let O1,O2 be Operation of X;
    assume
A1: for x holds x.O1 = x.O2;
    let a,b be object;
    thus [a,b] in O1 implies [a,b] in O2
    proof
      assume
A2:   [a,b] in O1;
      reconsider a,b as Element of X by A2,ZFMISC_1:87;
      b in a.O1 by A2,RELAT_1:169; then
      b in a.O2 by A1;
      hence thesis by RELAT_1:169;
    end;
    assume
A3: [a,b] in O2; then
A4: a in X & b in X by ZFMISC_1:87;
    reconsider a,b as Element of X by A3,ZFMISC_1:87;
    reconsider L = {a} as Subset of X by A4,ZFMISC_1:31;
    b in a.O2 by A3,RELAT_1:169; then
    b in a.O1 by A1;
    hence thesis by RELAT_1:169;
  end;
