reserve A for preIfWhileAlgebra,
  C,I,J for Element of A;
reserve S for non empty set,
  T for Subset of S,
  s for Element of S;

theorem Th75:
  for A being preIfWhileAlgebra st A is free for C,I,D,J being Element of A
  holds while(C,I) <> C & while(C,I) <> I &
  (while(C,I) = while(D,J) implies C = D & I = J)
proof
  let A be preIfWhileAlgebra such that
A1: A is free;
  let C,I,D,J be Element of A;
A2: dom Den(In(4, dom the charact of A), A) = 2-tuples_on the carrier of A
  by Th48;
A3: <*C,I*> in 2-tuples_on the carrier of A by FINSEQ_2:137;
A4: <*D,J*> in 2-tuples_on the carrier of A by FINSEQ_2:137;
A5: rng <*C,I*> = {C,I} by FINSEQ_2:127;
  then
A6: C in rng <*C,I*> by TARSKI:def 2;
  I in rng <*C,I*> by A5,TARSKI:def 2;
  hence while(C,I) <> C & while(C,I) <> I by A1,A2,A3,A6,Th38;
  assume while(C,I) = while(D,J);
  then <*C,I*> = <*D,J*> by A1,A2,A3,A4,Th36;
  hence thesis by FINSEQ_1:77;
end;
