reserve i,j,k,n for Nat;
reserve x,x1,x2,x3,y1,y2,y3 for set;

theorem
  [:{x1,x2,x3},{y1,y2,y3}:] = {[x1,y1],[x1,y2],[x1,y3],[x2,y1],[x2,y2],[
  x2,y3],[x3,y1],[x3,y2],[x3,y3]}
proof
  thus [:{x1,x2,x3},{y1,y2,y3}:] = [:{x1}\/{x2,x3}, {y1,y2,y3}:] by ENUMSET1:2
    .= [:{x1}\/{x2,x3}, {y1}\/ {y2,y3}:] by ENUMSET1:2
    .= [:{x1},{y1}:] \/ [:{x1},{y2,y3}:] \/ [:{x2,x3},{y1}:] \/ [:{x2,x3},{
  y2,y3}:] by ZFMISC_1:98
    .= [:{x1},{y1}:] \/ [:{x1},{y2,y3}:] \/ [:{x2,x3},{y1}:] \/ ([:{x2},{y2,
  y3}:] \/ [:{x3},{y2,y3}:]) by ZFMISC_1:109
    .= {[x1,y1]} \/ [:{x1},{y2,y3}:] \/ [:{x2,x3},{y1}:] \/ ([:{x2},{y2,y3}
  :] \/ [:{x3},{y2,y3}:]) by ZFMISC_1:29
    .= {[x1,y1]} \/ {[x1,y2],[x1,y3]} \/ [:{x2,x3},{y1}:] \/ ([:{x2},{y2,y3}
  :] \/ [:{x3},{y2,y3}:]) by ZFMISC_1:30
    .= {[x1,y1]} \/ {[x1,y2],[x1,y3]} \/ {[x2,y1],[x3,y1]} \/ ([:{x2},{y2,y3
  }:] \/ [:{x3},{y2,y3}:]) by ZFMISC_1:30
    .= {[x1,y1]} \/ {[x1,y2],[x1,y3]} \/ {[x2,y1],[x3,y1]} \/ ({[x2,y2],[x2,
  y3]} \/ [:{x3},{y2,y3}:]) by ZFMISC_1:30
    .= {[x1,y1]} \/ {[x1,y2],[x1,y3]} \/ {[x2,y1],[x3,y1]} \/ ({[x2,y2],[x2,
  y3]} \/ {[x3,y2],[x3,y3]}) by ZFMISC_1:30
    .= {[x1,y1],[x1,y2],[x1,y3]} \/ {[x2,y1],[x3,y1]} \/ ({[x2,y2],[x2,y3]}
  \/ {[x3,y2],[x3,y3]}) by ENUMSET1:2
    .= {[x1,y1],[x1,y2],[x1,y3],[x2,y1],[x3,y1]} \/ ({[x2,y2],[x2,y3]} \/ {[
  x3,y2],[x3,y3]}) by ENUMSET1:9
    .= {[x1,y1],[x1,y2],[x1,y3],[x2,y1],[x3,y1]} \/ {[x2,y2],[x2,y3],[x3,y2]
  ,[x3,y3]} by ENUMSET1:5
    .= {[x1,y1],[x1,y2],[x1,y3],[x2,y1],[x3,y1],[x2,y2],[x2,y3], [x3,y2],[x3
  ,y3]} by ENUMSET1:81
    .= {[x1,y1],[x1,y2],[x1,y3],[x2,y1]} \/ {[x3,y1],[x2,y2],[x2,y3],[x3,y2]
  ,[x3,y3]} by ENUMSET1:80
    .= {[x1,y1],[x1,y2],[x1,y3],[x2,y1]} \/ ({[x3,y1],[x2,y2],[x2,y3]} \/ {[
  x3,y2],[x3,y3]}) by ENUMSET1:9
    .= {[x1,y1],[x1,y2],[x1,y3],[x2,y1]} \/ ({[x2,y2],[x2,y3],[x3,y1]} \/ {[
  x3,y2],[x3,y3]}) by ENUMSET1:59
    .= {[x1,y1],[x1,y2],[x1,y3],[x2,y1]} \/ ({[x2,y2],[x2,y3],[x3,y1],[x3,y2
  ],[x3,y3]}) by ENUMSET1:9
    .= {[x1,y1],[x1,y2],[x1,y3],[x2,y1],[x2,y2],[x2,y3], [x3,y1],[x3,y2],[x3
  ,y3]} by ENUMSET1:80;
end;
