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;
reserve xx4 for Element of X4;

theorem
 for X1,X2,X3,X4 being non empty set
 for x being Element of [:X1,X2,X3,X4:]
  holds x <> x`1_4 & x <> x`2_4 & x <> x`3_4 & x <> x`4_4
proof let X1,X2,X3,X4 be non empty set;
  let x be Element of [:X1,X2,X3,X4:];
   reconsider Y = [:X1,X2:], X3, X4 as non empty set;
  reconsider x9=x as Element of [:Y,X3,X4:] by Th38;
  set Z9 = { x`1_4,x`2_4 }, Z = { Z9,{x`1_4}},
      Y9 = { Z,x`3_4 }, Y = { Y9,{Z} }, X9 =
  { Y,x`4_4 }, X = { X9,{Y} };
  x = [x`1_4,x`2_4,x`3_4,x`4_4]
    .= X;
  then
  x = x`1_4 or x = x`2_4
   implies X in Z9 & Z9 in Z & Z in Y9 & Y9 in Y & Y in X9 & X9 in X
     by TARSKI:def 2;
  hence x <> x`1_4 & x <> x`2_4 by XREGULAR:10;
  x`3_4 = (x qua set)`1`2
    .= x9`2_3;
  hence thesis by Th35;
end;
