
theorem
  for S, T being non empty reflexive RelStr st [:S,T:] is /\-complete
  holds S is /\-complete & T is /\-complete
proof
  let S, T be non empty reflexive RelStr such that
A1: for X being non empty Subset of [:S,T:] ex x being Element of [:S,T
:] st x is_<=_than X & for y being Element of [:S,T:] st y is_<=_than X holds x
  >= y;
A2: the carrier of [:S,T:] = [:the carrier of S,the carrier of T:] by
YELLOW_3:def 2;
  thus S is /\-complete
  proof
    set t = the Element of T;
    let X be non empty Subset of S;
    consider x being Element of [:S,T:] such that
A3: x is_<=_than [:X,{t}:] and
A4: for y being Element of [:S,T:] st y is_<=_than [:X,{t}:] holds x
    >= y by A1;
    take x`1;
A5: x = [x`1,x`2] by A2,MCART_1:21;
    hence x`1 is_<=_than X by A3,YELLOW_3:32;
    t <= t;
    then
A6: t is_<=_than {t} by YELLOW_0:7;
    let y be Element of S;
    assume y is_<=_than X;
    then x >= [y,t] by A4,A6,YELLOW_3:33;
    hence thesis by A5,YELLOW_3:11;
  end;
  set s = the Element of S;
  let X be non empty Subset of T;
  consider x being Element of [:S,T:] such that
A7: x is_<=_than [:{s},X:] and
A8: for y being Element of [:S,T:] st y is_<=_than [:{s},X:] holds x >=
  y by A1;
  take x`2;
A9: x = [x`1,x`2] by A2,MCART_1:21;
  hence x`2 is_<=_than X by A7,YELLOW_3:32;
  s <= s;
  then
A10: s is_<=_than {s} by YELLOW_0:7;
  let y be Element of T;
  assume y is_<=_than X;
  then x >= [s,y] by A8,A10,YELLOW_3:33;
  hence thesis by A9,YELLOW_3:11;
end;
