reserve a,b,c,k,m,n for Nat;
reserve p for Prime;

theorem Th15:
  for x,y being Nat st x = 36*k+14 & y = (12*k+5)*(18*k+7) holds
  x,y satisfy_Sierpinski_problem_35
  proof
    let x,y be Nat such that
A1: x = 36*k+14 and
A2: y = (12*k+5)*(18*k+7);
A3: x*(x+1) = 2*3*(12*k+5)*(18*k+7) = 6*y by A1,A2;
    y+1 = 6*(36*k*k+29*k+6) by A2;
    then 6 divides y+1;
    hence x*(x+1) divides y*(y+1) by A2,A3,INT_4:7;
A4: x = 2*(18*k+7) by A1;
    y = 2*(108*k*k+87*k+17)+1 by A2;
    hence not x divides y by A4;
A5: x+1 = 3*(12*k+5) by A1;
    then
A6: 3 divides x+1;
    y = 3*(72*k*k+58*k+11)+2 by A2;
    hence not x+1 divides y by A6,INT_2:9,NAT_4:9;
    0+7 <= 18*k+7 by XREAL_1:6;
    then
A7: 18*k+7 <> 1;
A8: 18*k+7 divides x by A4;
    18*k+7 divides y by A2;
    then not 18*k+7 divides y+1 by A7,NEWTON:39;
    hence not x divides y+1 by A8,INT_2:9;
    0+5 <= 12*k+5 by XREAL_1:6;
    then
A9: 12*k+5 <> 1;
A10: 12*k+5 divides x+1 by A5;
    12*k+5 divides y by A2;
    then not 12*k+5 divides y+1 by A9,NEWTON:39;
    hence not x+1 divides y+1 by A10,INT_2:9;
  end;
