reserve x,x1,x2,x3,x4,y,y1,y2,y3,y4,z,z1,z2,z2,z4 for object;
reserve X,X1,X2,X3,X4,Y for set;

theorem Th27:
 proj2(X \/ Y) = proj2 X \/ proj2 Y
proof
 thus proj2(X \/ Y) c= proj2 X \/ proj2 Y
  proof
   let y be object;
   assume y in proj2(X \/ Y);
    then consider x such that
A1: [x,y] in X \/ Y by Def13;
    [x,y] in X or [x,y] in Y by A1,XBOOLE_0:def 3;
    then y in proj2 X or y in proj2 Y by Def13;
   hence thesis by XBOOLE_0:def 3;
  end;
A2: proj2 Y c= proj2(X \/ Y) by Th9,XBOOLE_1:7;
  proj2 X c= proj2(X \/ Y) by Th9,XBOOLE_1:7;
 hence proj2 X \/ proj2 Y c= proj2(X \/ Y) by A2,XBOOLE_1:8;
end;
