 reserve n for Nat;
 reserve X,Y for Subset of TarskiEuclid2Space;

theorem
  for a being Element of TarskiEuclid2Space st
    (for x,y being Element of TarskiEuclid2Space st x in X & y in Y holds
      between a,x,y) & X is non empty & Y is non empty &
      (X is trivial implies X <> {a}) holds
        ex b being Element of TarskiEuclid2Space st
          X c= Line(Tn2TR a, Tn2TR b) & Y c= Line(Tn2TR a, Tn2TR b)
  proof
    let a be Element of TarskiEuclid2Space such that
A1: for x,y being Element of TarskiEuclid2Space st
      x in X & y in Y holds between a,x,y and
A1A: X is non empty and
A1B: Y is non empty and
A3: X is trivial implies X <> {a};
    consider x0 be object such that
K1: x0 in X by A1A;
    reconsider x0 as Element of TarskiEuclid2Space by K1;
    consider c be object such that
MM: c in Y by A1B;
    reconsider c as Element of TarskiEuclid2Space by MM;
V1: X c= LSeg(Tn2TR a,Tn2TR c)
    proof
      let x be object;
      assume
DA:   x in X;
      then reconsider x1 = x as Element of TarskiEuclid2Space;
      Tn2TR x1 in LSeg(Tn2TR a,Tn2TR c) by ThConv6,A1,DA,MM;
      hence x in LSeg(Tn2TR a,Tn2TR c);
    end;
t2: LSeg(Tn2TR a,Tn2TR c) c= Line(Tn2TR a,Tn2TR c) by RLTOPSP1:73; then
T1: X c= Line(Tn2TR a, Tn2TR c) by V1;
T2: x0 in Line(Tn2TR a, Tn2TR c) by t2,V1,K1;
    Y c= Line(Tn2TR a, Tn2TR c)
    proof
      let y be object;
      assume
V2:   y in Y; then
      reconsider y0 = y as Element of TarskiEuclid2Space;
      per cases;
      suppose
    MU: x0 = a;
        per cases;
        suppose X is trivial;
          then consider xx be object such that
      KL: X = {xx} by A1A,ZFMISC_1:131;
          thus thesis by MU,K1,KL,TARSKI:def 1,A3;
        end;
        suppose X is non trivial;
          then consider a0,b0 be object such that
     LO1: a0 in X and
     LO2: b0 in X and
     LO3: a0 <> b0;
          ex x1 be object st x1 in X & x1 <> a
          proof
            assume
        AA: for x1 be object holds not x1 in X or x1 = a;
            a0 <> a or b0 <> a by LO3;
            hence contradiction by LO1,LO2,AA;
          end; then
          consider x1 be object such that
      K1: x1 in X and
     VAL: x1 <> a;
          reconsider x1 as Element of TarskiEuclid2Space by K1;
N1:       Tn2TR x1 in LSeg(Tn2TR a,Tn2TR y0) by ThConv6,V2,K1,A1;
n2:       LSeg(Tn2TR a,Tn2TR y0) c= Line(Tn2TR a,Tn2TR y0) by RLTOPSP1:73;
          Tn2TR a in Line(Tn2TR a,Tn2TR y0) by RLTOPSP1:72; then
ff:       Line(Tn2TR x1,Tn2TR a) = Line(Tn2TR a,Tn2TR y0)
            by N1,n2,VAL,RLTOPSP1:75;
          Tn2TR a in Line(Tn2TR a,Tn2TR c) by RLTOPSP1:72;
          then Line(Tn2TR a,Tn2TR x1) c= Line(Tn2TR a,Tn2TR c)
            by K1,T1,RLTOPSP1:74;
          hence thesis by ff,RLTOPSP1:72;
        end;
      end;
      suppose
   VAL: x0 <> a;
    N1: Tn2TR x0 in LSeg(Tn2TR a,Tn2TR y0) by ThConv6,V2,K1,A1;
n2:     LSeg(Tn2TR a,Tn2TR y0) c= Line(Tn2TR a,Tn2TR y0) by RLTOPSP1:73;
        Tn2TR a in Line(Tn2TR a,Tn2TR y0) by RLTOPSP1:72; then
ff:     Line(Tn2TR x0,Tn2TR a) = Line(Tn2TR a,Tn2TR y0)
          by N1,n2,VAL,RLTOPSP1:75;
        Tn2TR a in Line(Tn2TR a,Tn2TR c) by RLTOPSP1:72;
        then Line(Tn2TR a,Tn2TR x0) c= Line(Tn2TR a,Tn2TR c) by T2,RLTOPSP1:74;
        hence thesis by ff,RLTOPSP1:72;
      end;
    end;
    hence thesis by T1;
  end;
