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;
reserve f for ExecutionFunction of A,S,T;

theorem Th88:
  for A being free preIfWhileAlgebra for I1,I2 being Element of A
  for n being Nat st I1\;I2 in (ElementaryInstructions A)|^n
  ex i being Nat st n = i+1 &
  I1 in (ElementaryInstructions A)|^i & I2 in (ElementaryInstructions A)|^i
proof
  let A be free preIfWhileAlgebra;
  set B = ElementaryInstructions A;
A1: B is GeneratorSet of A by Def25;
  let I1,I2 be Element of A;
A2: I1\;I2 <> I1 by Th73;
  I1\;I2 <> I2 by Th73;
  then I1\;I2 nin B by A2,Th50;
  then
A3: I1\;I2 nin B|^0 by Th18;
  let n be Nat;
  assume
A4: I1\;I2 in B|^n;
  then n > 0 by A3;
  then n >= 0+1 by NAT_1:13;
  then consider i being Nat such that
A5: n = 1+i by NAT_1:10;
  take i;
  thus n = i+1 by A5;
A6: dom Den(In(2, dom the charact of A), A) = 2-tuples_on the carrier of A
  by Th44;
A7: for o being OperSymbol of A, p being FinSequence st p in dom Den(o,A)
  holds Den(o,A).p in B implies o <> In(2, dom the charact of A)
  proof
    let o be OperSymbol of A;
    let p be FinSequence;
    assume that
A8: p in dom Den(o,A) and
A9: Den(o,A).p in B and
A10: o = In(2, dom the charact of A);
    consider a,b being object such that
A11: a in the carrier of A and
A12: b in the carrier of A and
A13: p = <*a,b*> by A6,A8,A10,FINSEQ_2:137;
    reconsider a,b as Element of A by A11,A12;
A14: a\;b <> a by Th73;
    a\;b <> b by Th73;
    hence contradiction by A9,A10,A13,A14,Th50;
  end;
  <*I1,I2*> in dom Den(In(2, dom the charact of A),A) by A6,FINSEQ_2:137;
  then rng <*I1,I2*> c= B|^i by A1,A4,A5,A7,Th39;
  then {I1,I2} c= B|^i by FINSEQ_2:127;
  hence thesis by ZFMISC_1:32;
end;
