reserve V, C for set;
reserve A, B, D for Element of Fin PFuncs (V, C);
reserve s for Element of PFuncs (V,C);

theorem Th7:
  for a be finite set holds a in B & (for b be finite set st b in B
  & b c= a holds b = a) implies a in mi B
proof
  let a be finite set;
  assume that
A1: a in B and
A2: for s be finite set st s in B & s c= a holds s = a;
  B c= PFuncs (V, C) by FINSUB_1:def 5;
  then reconsider a9 = a as Element of PFuncs (V, C) by A1;
  s in B & s c= a iff s = a by A1,A2;
  then
  a9 in { t where t is Element of PFuncs (V,C) : t is finite & for s being
  Element of PFuncs (V, C) holds s in B & s c= t iff s = t };
  hence thesis;
end;
