 reserve R for commutative Ring;
 reserve A,B for non degenerated commutative Ring;
 reserve h for Function of A,B;
 reserve I0,I,I1,I2 for Ideal of A;
 reserve J,J1,J2 for proper Ideal of A;
 reserve p for prime Ideal of A;
 reserve S,S1 for non empty Subset of A;
 reserve E,E1,E2 for Subset of A;
 reserve a,b,f for Element of A;
 reserve n for Nat;
 reserve x,o,o1 for object;
 reserve m for maximal Ideal of A;
 reserve p for prime Ideal of A;
 reserve k for non zero Nat;

theorem Th45:
  for P,Q being Point of ZariskiTS A st P <> Q holds
    ex V being Subset of ZariskiTS A
   st V is open & ( P in V & not Q in V or Q in V & not P in V )
   proof
     let P,Q be Point of ZariskiTS A;
     reconsider P1=P, Q1=Q as prime Ideal of A by Lm44;
     assume P <> Q; then
     per cases;
       suppose not Q c= P; then
         Q \ P <> {} by XBOOLE_1:37; then
         consider x such that
A3:      x in Q \ P by XBOOLE_0:def 1;
         x in Q1 by A3; then
         reconsider x as Element of A;
         PrimeIdeals(A,{x}-Ideal) is closed Subset of ZariskiTS A by Def7; then
         reconsider W = [#]ZariskiTS A \ PrimeIdeals(A,{x}-Ideal) as
           open Subset of ZariskiTS A by PRE_TOPC:def 3;
A5:      not x in P by XBOOLE_0:def 5,A3;
A6:      not P in PrimeIdeals(A,{x}-Ideal)
         proof
           assume P in PrimeIdeals(A,{x}-Ideal); then
           consider P0 be prime Ideal of A such that
A8:        P = P0 and
A9:        {x}-Ideal c= P0;
           thus contradiction by A5,IDEAL_1:66,A8,A9;
         end;
         {x}-Ideal c= Q1-Ideal by A3,ZFMISC_1:31,IDEAL_1:57; then
         {x}-Ideal c= Q1 by IDEAL_1:44; then
         Q in PrimeIdeals(A,{x}-Ideal); then
         P in W & not Q in W by A6,XBOOLE_0:def 5;
         hence thesis;
       end;
       suppose Q c= P & Q <> P; then
         Q c< P; then
         consider x such that
A13:     x in P and
A14:     not x in Q by XBOOLE_0:6;
         x in P1 by A13; then
         reconsider x as Element of A;
         PrimeIdeals(A,{x}-Ideal) is closed Subset of ZariskiTS A by Def7; then
         reconsider W = [#]ZariskiTS A \ PrimeIdeals(A,{x}-Ideal)
           as open Subset of ZariskiTS A by PRE_TOPC:def 3;
A16:     not Q in PrimeIdeals(A,{x}-Ideal)
         proof
           assume Q in PrimeIdeals(A,{x}-Ideal); then
           consider Q0 be prime Ideal of A such that
A18:       Q = Q0 and
A19:       {x}-Ideal c= Q0;
           thus contradiction by A14,IDEAL_1:66,A18,A19;
         end;
         {x}-Ideal c= P1-Ideal by A13,ZFMISC_1:31,IDEAL_1:57; then
         {x}-Ideal c= P1 by IDEAL_1:44; then
         P in PrimeIdeals(A,{x}-Ideal); then
         Q in W & not P in W by A16,XBOOLE_0:def 5;
         hence thesis;
       end;
     end;
