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 Th23:
 proj1(X \/ Y) = proj1 X \/ proj1 Y
proof
  thus proj1(X \/ Y) c= proj1 X \/ proj1 Y
  proof
   let x be object;
   assume x in proj1(X \/ Y);
    then consider y such that
A1: [x,y] in X \/ Y by Def12;
    [x,y] in X or [x,y] in Y by A1,XBOOLE_0:def 3;
    then x in proj1 X or x in proj1 Y by Def12;
   hence thesis by XBOOLE_0:def 3;
  end;
A2: proj1 Y c= proj1(X \/ Y) by Th8,XBOOLE_1:7;
   proj1 X c= proj1(X \/ Y) by Th8,XBOOLE_1:7;
  hence proj1 X \/ proj1 Y c= proj1 (X \/ Y) by A2,XBOOLE_1:8;
end;
