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 Th31:
 proj1_3(X \/ Y) = proj1_3 X \/ proj1_3 Y
proof
 thus proj1_3(X \/ Y) = proj1(proj1 X \/ proj1 Y) by Th23
     .= proj1_3 X \/ proj1_3 Y by Th23;
end;
