reserve A,B,C for Ordinal,
  K,L,M,N for Cardinal,
  x,y,y1,y2,z,u for object,X,Y,Z,Z1,Z2 for set,
  n for Nat,
  f,f1,g,h for Function,
  Q,R for Relation;
reserve ff for Cardinal-Function;

theorem Th16:
  pi(X \/ Y,x) = pi(X,x) \/ pi(Y,x)
proof
  thus pi(X \/ Y,x) c= pi(X,x) \/ pi(Y,x)
  proof
    let y be object;
    assume y in pi(X \/ Y,x);
    then consider f such that
A1: f in X \/ Y and
A2: y = f.x by Def6;
    f in X or f in Y by A1,XBOOLE_0:def 3;
    then y in pi(X,x) or y in pi(Y,x) by A2,Def6;
    hence thesis by XBOOLE_0:def 3;
  end;
  let y be object;
  assume y in pi(X,x) \/ pi(Y,x);
  then
A3: y in pi(X,x) or y in pi(Y,x) by XBOOLE_0:def 3;
A4: now
    assume y in pi(X,x);
    then consider f such that
A5: f in X and
A6: y = f.x by Def6;
    f in X \/ Y by A5,XBOOLE_0:def 3;
    hence thesis by A6,Def6;
  end;
  now
    assume not y in pi(X,x);
    then consider f such that
A7: f in Y and
A8: y = f.x by A3,Def6;
    f in X \/ Y by A7,XBOOLE_0:def 3;
    hence thesis by A8,Def6;
  end;
  hence thesis by A4;
end;
