reserve v,x,x1,x2,x3,x4,y,y1,y2,y3,y4,z,z1,z2 for object,
  X,X1,X2,X3,X4,Y,Y1,Y2,Y3,Y4,Y5,
  Z,Z1,Z2,Z3,Z4,Z5 for set;
reserve p for pair object;
reserve R for Relation;
reserve xx1 for Element of X1,
  xx2 for Element of X2,
  xx3 for Element of X3;

theorem
  [:{x1,y1},{x2,y2},{x3,y3}:] = { [x1,x2,x3],[x1,y2,x3],[x1,x2,y3],[x1,
  y2,y3], [y1,x2,x3],[y1,y2,x3],[y1,x2,y3],[y1,y2,y3] }
proof
A1: [:{[x1,x2],[x1,y2]},{x3,y3}:] = { [[x1,x2],x3],[[x1,x2],y3],[[x1,y2],x3]
  ,[[x1,y2],y3]} by Th17
    .= { [x1,x2,x3],[x1,y2,x3],[x1,x2,y3],[x1,y2,y3]} by ENUMSET1:62;
A2: [:{[y1,x2],[y1,y2]},{x3,y3}:] = { [[y1,x2],x3],[[y1,x2],y3],[[y1,y2],x3]
  ,[[y1,y2],y3]} by Th17
    .= { [y1,x2,x3],[y1,y2,x3],[y1,x2,y3],[y1,y2,y3]} by ENUMSET1:62;
  thus [:{x1,y1},{x2,y2},{x3,y3}:] = [:[:{x1,y1},{x2,y2}:],{x3,y3}:] by
ZFMISC_1:def 3
    .= [:{[x1,x2],[x1,y2],[y1,x2],[y1,y2]},{x3,y3}:] by Th17
    .= [:{[x1,x2],[x1,y2]} \/ {[y1,x2],[y1,y2]},{x3,y3}:] by ENUMSET1:5
    .= { [x1,x2,x3],[x1,y2,x3],[x1,x2,y3],[x1,y2,y3]} \/ {[y1,x2,x3],[y1,y2,
  x3],[y1,x2,y3],[y1,y2,y3] } by A1,A2,ZFMISC_1:97
    .= { [x1,x2,x3],[x1,y2,x3],[x1,x2,y3],[x1,y2,y3], [y1,x2,x3],[y1,y2,x3],
  [y1,x2,y3],[y1,y2,y3] } by ENUMSET1:25;
end;
