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 Th45:
  a in E & b in E & a <> b & not Collinear p,q,r & E = Plane(p,q,r) &
  not b in Line(p,q) & Line(p,q) <> Line(a,b) 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: not b in Line(p,q) and
A7: Line(p,q) <> Line(a,b);
    set A  = Line(p,q),
        A9 = Line(a,b);
      ex c be POINT of S st not c in A9 & c in A
      proof
        assume not (ex c be POINT of S st not c in A9 & c in A);
        then A9 is_line & p <> q & p in A & q in A & p in A9 & q in A9
          by A3,A4,GTARSKI3:46,83;
        hence thesis by A7,GTARSKI3:87;
      end;
    hence thesis by A1,A2,A3,A4,A5,A6,Th44;
  end;
