theorem :: GTARSKI1:45
  p <> q & s <> p & s in Line(p,q) implies Line(p,q) = Line(p,s)
  proof
    assume that
A1A: p <> q and
A1B: s <> p and
A1C: s in Line(p,q);
    consider x be POINT of S such that
A2: s = x & Collinear p,q,x by A1C;
    thus Line(p,q) c= Line(p,s)
    proof
      let t be object;
      assume t in Line(p,q);
      then consider y be POINT of S such that
A3:   t = y & Collinear p,q,y;
A4:   between p,q,y implies between p,s,y or between s,y,p or
        between y,p,s
      proof
        assume
A5:     between p,q,y;
A6:     between p,q,s implies between p,s,y or between s,y,p or
          between y,p,s
        proof
          assume between p,q,s;
          then between p,y,s or between p,s,y by A1A,A5,Satz5p1;
          hence thesis by Satz3p2;
        end;
A7:     between q,s,p implies between p,s,y or between s,y,p or
          between y,p,s
        proof
          assume between q,s,p;
          then between p,s,q by Satz3p2;
          hence thesis by A5,Satz3p6p2;
        end;
        between s,p,q implies between p,s,y or between s,y,p or
          between y,p,s
        proof
          assume between s,p,q;
          then between s,p,y by A1A,A5,Satz3p7p2;
          hence thesis by Satz3p2;
        end;
        hence thesis by A2,A6,A7;
      end;
A8:   between q,y,p implies between p,s,y or between s,y,p or
        between y,p,s
      proof
        assume between q,y,p;
        then
A9:     between p,y,q by Satz3p2;
A10:    between p,q,s implies between p,s,y or between s,y,p or
          between y,p,s
        proof
          assume between p,q,s;
          then between p,y,s by A9,Satz3p6p2;
          hence thesis by Satz3p2;
        end;
A11:    between q,s,p implies between p,s,y or between s,y,p or
          between y,p,s
        proof
          assume between q,s,p;
          then between p,s,q by Satz3p2;
          then between p,s,y or between p,y,s by A9,Satz5p3;
          hence thesis by Satz3p2;
        end;
        between s,p,q implies between p,s,y or between s,y,p or
          between y,p,s
        proof
          assume between s,p,q;
          then between s,p,y by A9,Satz3p5p1;
          hence thesis by Satz3p2;
        end;
        hence thesis by A2,A10,A11;
      end;
      between y,p,q implies between p,s,y or between s,y,p or
        between y,p,s
      proof
        assume
A12:    between y,p,q;
A13:    between q,s,p implies between p,s,y or between s,y,p or
          between y,p,s
        proof
          assume
A14:      between q,s,p;
          between q,p,y by A12,Satz3p2;
          then between s,p,y by A14,Satz3p6p1;
          hence thesis by Satz3p2;
        end;
        between s,p,q implies between p,s,y or between s,y,p or
          between y,p,s
        proof
          assume between s,p,q;
          then between q,p,s & between q,p,y by A12,Satz3p2;
          then between p,s,y or between p,y,s by A1A,Satz5p2;
          hence thesis by Satz3p2;
        end;
        hence thesis by A2,A12,A1A,Satz3p7p2,A13;
      end;
      then Collinear p,s,y by A4,A8,A3;
      hence thesis by A3;
    end;
    thus Line(p,s) c= Line(p,q)
    proof
      let t be object;
      assume t in Line(p,s);
      then consider y be POINT of S such that
A15:  t = y & Collinear p,s,y;
A16:  between p,s,y implies between p,q,y or between q,y,p or between y,p,q
      proof
        assume
A17:    between p,s,y;
A18:    between q,s,p implies between p,q,y or between q,y,p  or
          between y,p,q
        proof
          assume between q,s,p;
          then between p,s,q by Satz3p2;
          then between p,q,y or between p,y,q by A17,A1B,Satz5p1;
          hence thesis by Satz3p2;
        end;
        between s,p,q implies between p,q,y or between q,y,p or
          between y,p,q
        proof
          assume between s,p,q;
          then between q,p,s by Satz3p2;
          then between q,p,y by A1B,A17,Satz3p7p2;
          hence thesis by Satz3p2;
        end;
        hence thesis by A17,Satz3p6p2,A18,A2;
      end;
A19:  between s,y,p implies between p,q,y or between q,y,p or between y,p,q
      proof
        assume
A20:    between s,y,p;
A21:    between p,q,s implies between p,q,y or between q,y,p or
          between y,p,q
        proof
          assume
A22:      between p,q,s;
          between p,y,s by A20,Satz3p2;
          then between p,q,y or between p,y,q by A22,Satz5p3;
          hence thesis by Satz3p2;
        end;
A23:    between q,s,p implies between p,q,y or between q,y,p  or
          between y,p,q
        proof
          assume between q,s,p;
          then
A25:      between p,s,q by Satz3p2;
          between p,y,s by A20,Satz3p2;
          then between p,y,q by A25,Satz3p6p2;
          hence thesis by Satz3p2;
        end;
        between s,p,q implies between p,q,y or between q,y,p or
          between y,p,q
        proof
          assume between s,p,q;
          then
A26:      between q,p,s by Satz3p2;
          between p,y,s by A20,Satz3p2;
          then between q,p,y by A26,Satz3p5p1;
          hence thesis by Satz3p2;
        end;
        hence thesis by A21,A23,A2;
      end;
      between y,p,s implies between p,q,y or between q,y,p or between y,p,q
      proof
        assume
A27:    between y,p,s;
A28:    between q,s,p implies between p,q,y or between q,y,p  or
          between y,p,q
        proof
          assume between q,s,p;
          then between p,s,q by Satz3p2;
          hence thesis by A27,A1B,Satz3p7p2;
        end;
        between s,p,q implies between p,q,y or between q,y,p or
          between y,p,q
        proof
          assume
A29:      between s,p,q;
          between s,p,y by A27,Satz3p2;
          then between p,q,y or between p,y,q by Satz5p2,A29,A1B;
          hence thesis by Satz3p2;
        end;
        hence thesis by A27,Satz3p5p1,A28,A2;
      end;
      then Collinear p,q,y by A15,A16,A19;
      hence thesis by A15;
    end;
  end;
