reserve A,B,X,X1,Y,Y1,Y2,Z for set, a,x,y,z for object;
reserve P,R for Relation of X,Y;
reserve D,D1,D2,E,F for non empty set;
reserve R for Relation of D,E;
reserve x for Element of D;
reserve y for Element of E;

theorem
  for N being set, R,S being Relation of N st for i being set st i in N
  holds Im(R,i) = Im(S,i) holds R = S
proof
  let N be set, R,S be Relation of N such that
A1: for i being set st i in N holds Im(R,i) = Im(S,i);
  let a,b be Element of N;
  thus [a,b] in R implies [a,b] in S
  proof
    assume
A2: [a,b] in R;
    then
A3: a in dom R by XTUPLE_0:def 12;
    a in {a} by TARSKI:def 1;
    then b in Im(R,a) by A2,RELAT_1:def 13;
    then b in Im(S,a) by A1,A3;
    then ex e being object st [e,b] in S & e in {a} by RELAT_1:def 13;
    hence thesis by TARSKI:def 1;
  end;
  assume
A4: [a,b] in S;
  then
A5: a in dom S by XTUPLE_0:def 12;
  a in {a} by TARSKI:def 1;
  then b in Im(S,a) by A4,RELAT_1:def 13;
  then b in Im(R,a) by A1,A5;
  then ex e being object st [e,b] in R & e in {a} by RELAT_1:def 13;
  hence thesis by TARSKI:def 1;
end;
