reserve A, B for non empty preBoolean set,
  x, y for Element of [:A,B:];
reserve X for set,
  a,b,c for Element of [:A,B:];
reserve a for Element of [:Fin X, Fin X:];
reserve A for set;

theorem
  for X being non empty set for f being Function of X, [:Fin A,Fin A:]
  for B being Element of Fin X for x being Element of X st x in B holds f.x c=
  FinPairUnion(B,f)
proof
  let X be non empty set, f be Function of X, [:Fin A,Fin A:];
  let B be (Element of Fin X), x be Element of X;
  assume
A1: x in B;
  then FinPairUnion A $$(B, f) = FinPairUnion A $$(B \/ {.x.}, f) by
ZFMISC_1:40
    .= FinPairUnion A.(FinPairUnion A $$(B,f),f.x) by A1,SETWISEO:20
    .= FinPairUnion A $$(B,f) \/ f.x by Def6;
  hence thesis by Th10;
end;
