
theorem
  for X, Y being non empty reflexive RelStr st [:X,Y:] is with_infima
  holds X is with_infima & Y is with_infima
proof
  let X, Y be non empty reflexive RelStr such that
A1: [:X,Y:] is with_infima;
A2: the carrier of [:X,Y:] = [:the carrier of X, the carrier of Y:] by Def2;
  thus X is with_infima
  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 thesis 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 thesis by A10,Th11;
end;
