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, B being Element of Fin X for f,g being
  Function of X,[:Fin A,Fin A:] st f|B = g|B holds FinPairUnion(B,f) =
  FinPairUnion(B,g)
proof
  let X be non empty set, B be Element of Fin X;
  let f,g be Function of X,[:Fin A,Fin A:];
  set J = FinPairUnion A;
A1: [{}.A, {}.A] = the_unity_wrt J by Th19;
  assume
A2: f|B = g|B;
  now
    per cases;
    suppose
A3:   B = {};
      hence FinPairUnion(B,f) = J$$({}.X,f)
        .= [{}.A, {}.A] by A1,Th18,SETWISEO:31
        .= J$$({}.X,g) by A1,Th18,SETWISEO:31
        .= FinPairUnion(B,g) by A3;
    end;
    suppose
A4:   B <> {};
      f.:B = g.:B by A2,RELAT_1:166;
      hence thesis by A4,SETWISEO:26;
    end;
  end;
  hence thesis;
end;
