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 Th72:
  for A being preIfWhileAlgebra st A is free for C,I1,I2 being Element of A
  holds EmptyIns A <> I1\;I2 & EmptyIns A <> if-then-else(C,I1,I2) &
  EmptyIns A <> while(C,I1)
proof
  let A be preIfWhileAlgebra such that
A1: A is free;
  let C,I1,I2 be Element of A;
A2: dom Den(In(1, dom the charact of A), A) = {{}} by Th42;
A3: dom Den(In(2, dom the charact of A), A) = 2-tuples_on the carrier of A
  by Th44;
A4: dom Den(In(3, dom the charact of A), A) = 3-tuples_on the carrier of A
  by Th47;
A5: dom Den(In(4, dom the charact of A), A) = 2-tuples_on the carrier of A
  by Th48;
A6: {} in {{}} by TARSKI:def 1;
  <*I1,I2*> in 2-tuples_on the carrier of A by FINSEQ_2:137;
  hence EmptyIns A <> I1\;I2 by A1,A2,A3,A6,Th36;
  <*C,I1,I2*> in 3-tuples_on the carrier of A by FINSEQ_2:139;
  hence EmptyIns A <> if-then-else(C,I1,I2) by A1,A2,A4,A6,Th36;
  <*C,I1*> in 2-tuples_on the carrier of A by FINSEQ_2:137;
  hence thesis by A1,A2,A5,A6,Th36;
end;
