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 Th73:
  for A being preIfWhileAlgebra st A is free
  for I1,I2,C,J1,J2 being Element of A holds I1\;I2 <> I1 & I1\;I2 <> I2 &
  (I1\;I2 = J1\;J2 implies I1 = J1 & I2 = J2) &
  I1\;I2 <> if-then-else(C,J1,J2) & I1\;I2 <> while(C,J1)
proof
  let A be preIfWhileAlgebra such that
A1: A is free;
  let I1,I2,C,J1,J2 be Element of A;
A2: 2 in dom the charact of A by Def11;
A3: dom Den(In(2, dom the charact of A), A) = 2-tuples_on the carrier of A
  by Th44;
A4: In(2, dom the charact of A) = 2 by A2,SUBSET_1:def 8;
A5: 3 in dom the charact of A by Def12;
A6: dom Den(In(3, dom the charact of A), A) = 3-tuples_on the carrier of A
  by Th47;
A7: In(3, dom the charact of A) = 3 by A5,SUBSET_1:def 8;
A8: 4 in dom the charact of A by Def13;
A9: dom Den(In(4, dom the charact of A), A) = 2-tuples_on the carrier of A
  by Th48;
A10: In(4, dom the charact of A) = 4 by A8,SUBSET_1:def 8;
A11: <*I1,I2*> in 2-tuples_on the carrier of A by FINSEQ_2:137;
A12: <*J1,J2*> in 2-tuples_on the carrier of A by FINSEQ_2:137;
A13: rng <*I1,I2*> = {I1,I2} by FINSEQ_2:127;
  then
A14: I1 in rng <*I1,I2*> by TARSKI:def 2;
  I2 in rng <*I1,I2*> by A13,TARSKI:def 2;
  hence I1\;I2 <> I1 & I1\;I2 <> I2 by A1,A3,A11,A14,Th38;
  hereby
    assume I1\;I2 = J1\;J2;
    then <*I1,I2*> = <*J1,J2*> by A1,A3,A11,A12,Th36;
    hence I1 = J1 & I2 = J2 by FINSEQ_1:77;
  end;
  <*C,J1,J2*> in 3-tuples_on the carrier of A by FINSEQ_2:139;
  hence I1\;I2 <> if-then-else(C,J1,J2) by A1,A3,A4,A6,A7,A11,Th36;
  <*C,J1*> in 2-tuples_on the carrier of A by FINSEQ_2:137;
  hence thesis by A1,A3,A4,A9,A10,A11,Th36;
end;
