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
  for A being Universal_Algebra for B being Subset of A
  holds Constants A c= B|^1
proof
  let A be Universal_Algebra;
  let B be Subset of A;
  let x be object;
  assume x in Constants A;
  then consider a being Element of A such that
A1: x = a and
A2: ex o being Element of Operations A st arity o = 0 & a in rng o;
  consider o being Element of Operations A such that
A3: arity o = 0 and
A4: a in rng o by A2;
  consider s being object such that
A5: s in dom o and
A6: a = o.s by A4,FUNCT_1:def 3;
  consider z being object such that
A7: z in dom the charact of A and
A8: o = (the charact of A).z by FUNCT_1:def 3;
  reconsider z as Element of dom the charact of A by A7;
A9: Den(z,A) = o by A8;
A10: s is Element of 0-tuples_on the carrier of A by A3,A5,MARGREL1:22;
  reconsider s as Element of (the carrier of A)* by A5;
  rng s c= B by A10;
  then
A11: x in {Den(r,A).p where r is (Element of dom the charact of A),
  p is Element of (the carrier of A)*: p in dom Den(r,A) & rng p c= B}
  by A1,A5,A6,A9;
  B|^0 = B by Th18;
  then B|^(0+1) = B \/ {Den(r,A).p where r is (Element of dom the charact of
  A), p is Element of (the carrier of A)*: p in dom Den(r,A) & rng p c= B}
  by Th19;
  hence thesis by A11,XBOOLE_0:def 3;
end;
