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 Th68:
  for X being disjoint_with_NAT non empty set
  for I being Element of FreeUnivAlgNSG(ECIW-signature,X)
  st I in ElementaryInstructions FreeUnivAlgNSG(ECIW-signature,X)
  ex x being Element of X st I = x-tree {}
proof
  let X be disjoint_with_NAT non empty set;
  set S = ECIW-signature;
  set A = FreeUnivAlgNSG(S, X);
  let I be Element of FreeUnivAlgNSG(ECIW-signature,X) such that
A1: I in ElementaryInstructions FreeUnivAlgNSG(ECIW-signature,X);
  per cases by Th56;
  suppose ex x being Element of X st I = root-tree x;
    then consider x being Element of X such that
A2: I = root-tree x;
    root-tree x = x-tree {} by TREES_4:20;
    hence thesis by A2;
  end;
  suppose
    ex n being Nat, p being FinSequence of FreeUnivAlgNSG(ECIW-signature,X)
    st n in Seg 4 & I = n-tree p & len p = S.n;
    then consider n being Nat,
    p being FinSequence of FreeUnivAlgNSG(ECIW-signature,X) such that
A3: n in Seg 4 and
A4: I = n-tree p and len p = S.n;
    per cases by A3,ENUMSET1:def 2,FINSEQ_3:2;
    suppose
A5:   n = 1;
      then p = {} by A4,Th58;
      then I = EmptyIns A by A4,A5,Th57;
      hence thesis by A1,Th49;
    end;
    suppose
A6:   n = 2;
      then consider I1,I2 being Element of A such that
A7:   p = <*I1,I2*> by A4,Th60;
A8:   I = n-tree(I1,I2) by A4,A7
        .= I1\;I2 by A6,Th59;
      then
A9:   I <> I1 by Th61;
      I <> I2 by A8,Th61;
      hence thesis by A1,A8,A9,Th50;
    end;
    suppose
A10:  n = 3;
      then consider C,I1,I2 being Element of A such that
A11:  p = <*C,I1,I2*> by A4,Th64;
      I = if-then-else(C,I1,I2) by A4,A10,A11,Th63;
      hence thesis by A1,Th51;
    end;
    suppose
A12:  n = 4;
      then consider C,I9 being Element of A such that
A13:  p = <*C,I9*> by A4,Th67;
      I = n-tree(C,I9) by A4,A13
        .= while(C,I9) by A12,Th66;
      hence thesis by A1,Th52;
    end;
  end;
end;
