
theorem
  for X, Y being non empty reflexive RelStr st [:X,Y:] is with_suprema
  holds X is with_suprema & Y is with_suprema
proof
  let X, Y be non empty reflexive RelStr such that
A1: [:X,Y:] is with_suprema;
A2: the carrier of [:X,Y:] = [:the carrier of X, the carrier of Y:] by Def2;
  thus X is with_suprema
  proof
    let x, y be Element of X;
    set a = the Element of Y;
A3: a <= a;
    consider z being Element of [:X,Y:] such that
A4: [x,a] <= z & [y,a] <= z and
A5: for z9 being Element of [:X,Y:] st [x,a] <= z9 & [y,a] <= z9 holds
    z <= z9 by A1;
    take z`1;
A6: z = [z`1,z`2] by A2,MCART_1:21;
    hence x <= z`1 & y <= z`1 by A4,Th11;
    let z9 be Element of X;
    assume x <= z9 & y <= z9;
    then [x,a] <= [z9,a] & [y,a] <= [z9,a] by A3,Th11;
    then z <= [z9,a] by A5;
    hence z`1 <= z9 by A6,Th11;
  end;
  set a = the Element of X;
A7: a <= a;
  let x, y be Element of Y;
  consider z being Element of [:X,Y:] such that
A8: [a,x] <= z & [a,y] <= z and
A9: for z9 being Element of [:X,Y:] st [a,x] <= z9 & [a,y] <= z9 holds
  z <= z9 by A1;
  take z`2;
A10: z = [z`1,z`2] by A2,MCART_1:21;
  hence x <= z`2 & y <= z`2 by A8,Th11;
  let z9 be Element of Y;
  assume x <= z9 & y <= z9;
  then [a,x] <= [a,z9] & [a,y] <= [a,z9] by A7,Th11;
  then z <= [a,z9] by A9;
  hence z`2 <= z9 by A10,Th11;
end;
