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 Th71:
  for A being preIfWhileAlgebra holds Generators A c= ElementaryInstructions A
proof
  let A be preIfWhileAlgebra;
  let x be object;
  assume that
A1: x in Generators A and
A2: x nin ElementaryInstructions A;
  reconsider x as Element of A by A1;
  dom Den(In(1, dom the charact of A), A) = {{}} by Th42;
  then
A3: {} in dom Den(In(1, dom the charact of A),A) by TARSKI:def 1;
  per cases by A2,Th53;
  suppose x = EmptyIns A;
    then x in rng Den(In(1, dom the charact of A), A) by A3,FUNCT_1:3;
    hence contradiction by A1,Th26;
  end;
  suppose
    ex I1,I2 being Element of A st x = I1\;I2 & I1 <> I1\;I2 & I2 <> I1\;I2;
    then consider I1,I2 being Element of A such that
A4: x = I1\;I2 and I1 <> I1\;I2
    and I2 <> I1\;I2;
    dom Den(In(2, dom the charact of A), A) = 2-tuples_on the carrier of A
    by Th44;
    then <*I1,I2*> in dom Den(In(2, dom the charact of A),A) by FINSEQ_2:137;
    then x in rng Den(In(2, dom the charact of A),A) by A4,FUNCT_1:3;
    hence contradiction by A1,Th26;
  end;
  suppose ex C,I1,I2 being Element of A st x = if-then-else(C,I1,I2);
    then consider C,I1,I2 being Element of A such that
A5: x = if-then-else(C,I1,I2);
    dom Den(In(3, dom the charact of A), A) = 3-tuples_on the carrier of A
    by Th47;
    then <*C,I1,I2*> in dom Den(In(3, dom the charact of A),A) by FINSEQ_2:139;
    then x in rng Den(In(3, dom the charact of A),A) by A5,FUNCT_1:3;
    hence contradiction by A1,Th26;
  end;
  suppose ex C,J being Element of A st x = while(C,J);
    then consider C,J being Element of A such that
A6: x = while(C,J);
    dom Den(In(4, dom the charact of A), A) = 2-tuples_on the carrier of A
    by Th48;
    then <*C,J*> in dom Den(In(4, dom the charact of A),A) by FINSEQ_2:137;
    then x in rng Den(In(4, dom the charact of A),A) by A6,FUNCT_1:3;
    hence contradiction by A1,Th26;
  end;
end;
