reserve C1, C2 for Coherence_Space;

theorem
  for x1,y1, x2,y2 being set holds [[x1,x2],[y1,y2]] in Web (C1 [*] C2)
  iff [x1,y1] in Web C1 & [x2,y2] in Web C2
proof
  let x1,y1, x2,y2 be set;
A1: {[x1,x2],[y1,y2]} c= [:{x1,y1}, {x2,y2}:]
  proof
    let x,y be object;
    assume [x,y] in {[x1,x2],[y1,y2]};
    then
    [x,y] = [x1,x2] & x1 in {x1,y1} & x2 in {x2,y2}
     or [x,y] = [y1,y2] & y1 in {x1
    ,y1} & y2 in {x2,y2} by TARSKI:def 2;
    hence thesis by ZFMISC_1:87;
  end;
A2: proj1 {[x1,x2],[y1,y2]} = {x1,y1} & proj2 {[x1,x2],[y1,y2]} = {x2,y2} by
FUNCT_5:13;
  hereby
    assume [[x1,x2],[y1,y2]] in Web (C1 [*] C2);
    then {[x1,x2],[y1,y2]} in C1 [*] C2 by COH_SP:5;
    then {x1,y1} in C1 & {x2,y2} in C2 by A2,Th97;
    hence [x1,y1] in Web C1 & [x2,y2] in Web C2 by COH_SP:5;
  end;
  assume [x1,y1] in Web C1 & [x2,y2] in Web C2;
  then {x1,y1} in C1 & {x2,y2} in C2 by COH_SP:5;
  then {[x1,x2],[y1,y2]} in C1 [*] C2 by A1,Th96;
  hence thesis by COH_SP:5;
end;
