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 Th63:
  for X being disjoint_with_NAT non empty set
  for C,I1,I2 being Element of FreeUnivAlgNSG(ECIW-signature,X)
  holds if-then-else(C,I1,I2) = 3-tree<*C,I1,I2*>
proof
  let X be disjoint_with_NAT non empty set;
  set S = ECIW-signature;
  reconsider s = S as non empty FinSequence of omega;
  set A = FreeUnivAlgNSG(S,X);
  let C,I1,I2 be Element of A;
A1: 3 in dom the charact of A by Def12;
  reconsider f = (the charact of A).3 as 3-ary non empty homogeneous
  quasi_total PartFunc of (the carrier of A)*, the carrier of A by Def12;
A2: f = FreeOpNSG(3,S,X) by A1,FREEALG:def 11;
A3: 3 in dom S by Th54;
  then s/.3 = S.3 by PARTFUN1:def 6;
  then
A4: dom FreeOpNSG(3,S,X) = 3-tuples_on TS(DTConUA(S,X))
  by A3,Th54,FREEALG:def 10;
A5: <*C,I1,I2*> in 3-tuples_on TS(DTConUA(S,X)) by FINSEQ_2:139;
  thus if-then-else(C,I1,I2) = f.<*C,I1,I2*> by A1,SUBSET_1:def 8
    .= Sym(3,S,X)-tree(<*C,I1,I2*>) by A2,A3,A4,A5,FREEALG:def 10
    .= 3-tree <*C,I1,I2*> by A3,FREEALG:def 9;
end;
