reserve v,x,x1,x2,x3,x4,y,y1,y2,y3,y4,z,z1,z2 for object,
  X,X1,X2,X3,X4,Y,Y1,Y2,Y3,Y4,Y5,
  Z,Z1,Z2,Z3,Z4,Z5 for set;
reserve p for pair object;
reserve R for Relation;

theorem Th18:
  X <> {} & Y <> {} implies for x being Element of [:X,Y:] holds x
  <> x`1 & x <> x`2
proof
  assume
A1: X <> {} & Y <> {};
  let x be Element of [:X,Y:];
  x = [x`1,x`2] by A1,Th15;
  hence thesis by Th14;
end;
