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 Th74:
  for A being preIfWhileAlgebra st A is free
  for C,I1,I2,D,J1,J2 being Element of A
  holds if-then-else(C,I1,I2) <> C & if-then-else(C,I1,I2) <> I1 &
  if-then-else(C,I1,I2) <> I2 & if-then-else(C,I1,I2) <> while(D,J1) &
  (if-then-else(C,I1,I2) = if-then-else(D,J1,J2) implies C=D & I1=J1 & I2=J2)
proof
  let A be preIfWhileAlgebra such that
A1: A is free;
  let C,I1,I2,D,J1,J2 be Element of A;
A2: 3 in dom the charact of A by Def12;
A3: dom Den(In(3, dom the charact of A), A) = 3-tuples_on the carrier of A
  by Th47;
A4: In(3, dom the charact of A) = 3 by A2,SUBSET_1:def 8;
A5: 4 in dom the charact of A by Def13;
A6: dom Den(In(4, dom the charact of A), A) = 2-tuples_on the carrier of A
  by Th48;
A7: In(4, dom the charact of A) = 4 by A5,SUBSET_1:def 8;
A8: <*C,I1,I2*> in 3-tuples_on the carrier of A by FINSEQ_2:139;
A9: <*D,J1,J2*> in 3-tuples_on the carrier of A by FINSEQ_2:139;
A10: rng <*C,I1,I2*> = {C,I1,I2} by FINSEQ_2:128;
  then
A11: C in rng <*C,I1,I2*> by ENUMSET1:def 1;
A12: I1 in rng <*C,I1,I2*> by A10,ENUMSET1:def 1;
  I2 in rng <*C,I1,I2*> by A10,ENUMSET1:def 1;
  hence if-then-else(C,I1,I2) <> C & if-then-else(C,I1,I2) <> I1 &
  if-then-else(C,I1,I2) <> I2 by A1,A3,A8,A11,A12,Th38;
  <*D,J1*> in 2-tuples_on the carrier of A by FINSEQ_2:137;
  hence if-then-else(C,I1,I2) <> while(D,J1) by A1,A3,A4,A6,A7,A8,Th36;
  assume if-then-else(C,I1,I2) = if-then-else(D,J1,J2);
  then <*C,I1,I2*> = <*D,J1,J2*> by A1,A3,A8,A9,Th36;
  hence thesis by FINSEQ_1:78;
end;
