reserve T, X, Y for Subset of MC-wff;
reserve p, q, r, s for Element of MC-wff;
reserve T, X, Y for Subset of MC-wff;
reserve p, q, r for Element of MC-wff;
reserve T, X, Y for Subset of MC-wff;
reserve p, q, r for Element of MC-wff;

theorem Th80:
  CnCPC (X) c= CnS4 (X)
proof
  let a be object;
  assume
A1: a in CnCPC (X);
  then reconsider r = a as Element of MC-wff;
  for T st T is S4_theory & X c= T holds r in T
  proof
    let T such that
A2: T is S4_theory and
A3: X c= T;
    T is CPC_theory by A2;
    hence thesis by A1,A3,Def20;
  end;
  hence thesis by Def23;
end;
