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 Th35:
 for X1,X2,X3 being non empty set
 for x being Element of [:X1,X2,X3:]
   holds x <> x`1_3 & x <> x`2_3 & x <> x`3_3
proof let X1,X2,X3 be non empty set;
  let x be Element of [:X1,X2,X3:];
  set Y9 = { x`1_3,x`2_3 }, Y = { Y9,{x`1_3}},
      X9 = { Y,x`3_3 }, X = { X9,{Y} };
A1: x = [x`1_3,x`2_3,x`3_3]
    .= [[x`1_3,x`2_3],x`3_3];
  then x = x`1_3 or x = x`2_3 implies X in Y9 & Y9 in Y & Y in X9 & X9 in X
    by TARSKI:def 2;
  hence x <> x`1_3 & x <> x`2_3 by XREGULAR:8;
  thus thesis by A1,Th14;
end;
