reserve S for non empty satisfying_CongruenceIdentity
              satisfying_SegmentConstruction
              satisfying_BetweennessIdentity
              satisfying_Pasch
              TarskiGeometryStruct;
reserve a,b for POINT of S;
reserve A for Subset of S;
reserve S for non empty satisfying_Tarski-model
              TarskiGeometryStruct;
reserve a,b,c,m,r,s for POINT of S;
reserve A for Subset of S;
reserve S         for non empty satisfying_Lower_Dimension_Axiom
                                satisfying_Tarski-model
                                TarskiGeometryStruct,
        a,b,c,d,m,p,q,r,s,x for POINT of S,
        A,A9,E              for Subset of S;

theorem Th44:
  a in E & b in E & a <> b & not Collinear p,q,r & E = Plane(p,q,r) &
  c in Line(p,q) & not c in Line(a,b) & not b in Line(p,q) implies
  Line(a,b) c= E & ex c st not Collinear a,b,c & E = Plane(a,b,c)
  proof
    assume that
A1: a in E and
A2: b in E and
A3: a <> b and
A4: not Collinear p,q,r and
A5: E = Plane(p,q,r) and
A6: c in Line(p,q) and
A7: not c in Line(a,b) and
A8: not b in Line(p,q);
    set  A  = Line(p,q), A9 = Line(a,b), A99 = Line(c,b);
A9: not r in A
    proof
      assume r in A;
      then ex s be POINT of S st s = r & Collinear p,q,s;
      hence thesis by A4;
    end;
    per cases;
    suppose c = b;
      then Collinear a,b,c by GTARSKI4:4;
      hence thesis by A7;
    end;
    suppose
A10:  c <> b;
A11:  p <> q by A4,GTARSKI3:46;
      then
A12:  A is_line & A9 is_line & A99 is_line by A10,A3;
A13:  A <> A99 by A8,GTARSKI3:83;
A14:  A,A99 Is c by A6,A8,A12,GTARSKI3:83;
A15:  not r in A & not b in A & b in Plane(A,r) by A9,A8,A2,A4,A5,Def11;
        T1: A is_line & A99 is_line by A11,A10;
        c in A & c in A99 by A6,GTARSKI3:83;
        then A /\ A99 is non empty by XBOOLE_0:def 4;
       then consider r99 be POINT of S such that
A16:   not r99 in A and r99 in A99 and
A17:   Plane(A,A99) = Plane(A,r99) by T1,A13,Def13;
         e2: A99 c= Plane(A,A99) & b in A99 by A14,Th43,GTARSKI3:83;
A19:   E = Plane(A,r) by A4,A5,Def11
        .= Plane(A,b) by A12,A15,Th34
        .= Plane(A,A99) by A8,A16,e2,A17,A12,Th34;
        W1: A is_line & A99 is_line by A11,A10;
         c in A & c in A99 by A6,GTARSKI3:83;
         then A99 /\ A is non empty by XBOOLE_0:def 4;
       then consider r999 be POINT of S such that
A20:   not r999 in A99 and r999 in A and
A21:   Plane(A99,A) = Plane(A99,r999) by W1,A13,Def13;
A22:    not a in A99
         proof
           assume
A23:       a in A99;
           b in A99 by GTARSKI3:83;
           then A99 = Line(a,b) by A3,A23,A12,GTARSKI3:87;
           hence thesis by A7,GTARSKI3:83;
         end;
         T23: a in Plane(A99,r999) by A1,A19,A14,A21,Th43;
A24:   a in A9 by GTARSKI3:83;
A25:     A9 <> A99 by GTARSKI3:83,A22;
         b in A9 & b in A99 by GTARSKI3:83;
         then W1: A99 /\ A9 is non empty by XBOOLE_0:def 4;
A26:   A9,A99 Is b by A12,A22,GTARSKI3:83;
       consider s be POINT of S such that
A27:   not s in A9 and s in A99 and
A28:   Plane(A9,A99) = Plane(A9,s) by A25,W1,A12,Def13;
         J1:A9 is_line by A3;
         j3:A99 c= Plane(A9,A99) & c in A99 by A26,Th43,GTARSKI3:83;
A30:   E = Plane(A99,A) by A19,A14,Th43
        .= Plane(A99,a) by A21,A22,T1,
      A20,T23,Th34
        .= Plane(A99,A9) by A22,W1,A24,A12,Def13
        .= Plane(A9,A99) by A26,Th43
        .= Plane(Line(a,b),c) by J1,A27,j3,A7,A28,Th34;
       thus Line(a,b) c= E by A12,A7,A30,Th31;
A31:   not Collinear a,b,c by A7;
       take c;
       thus thesis by A31,A30,Def11;
     end;
   end;
