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 Th46:
  E is_plane & a in E & b in E & 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: E is_plane and
A2: a in E and
A3: b in E and
A4: a <> b;
    consider p,q,r be POINT of S such that
A5: not Collinear p,q,r and
A6: E = Plane(p,q,r) by A1;
A7: p <> q by A5,GTARSKI3:46;
    set A  = Line(p,q),
        A9 = Line(a,b);
    per cases;
    suppose
A8:   A = A9;
      then
A9:   E = Plane(Line(a,b),r) by A5,A6,Def11;
A10:  not r in Line(a,b)
      proof
        assume r in Line(a,b);
        then ex s be POINT of S st r = s & Collinear p,q,s by A8;
        hence thesis by A5;
      end;
      Line(a,b) is_line by A4;
      hence Line(a,b) c= E by A9,A10,Th31;
      R1:not Collinear a,b,r by A10;
        then Plane(a,b,r) = Plane(Line(a,b),r) by Def11;
      hence ex c st not Collinear a,b,c & E = Plane(a,b,c)
        by R1,A8,A5,A6,Def11;
    end;
    suppose
A11:  A <> A9;
      not (a in A & b in A)
      proof
        assume
A12:    a in A & b in A;
        A is_line by A7;
        hence contradiction by A11,A12,A4,GTARSKI3:87;
      end;
      then per cases;
      suppose
A13:    not a in A;
        then consider c such that
A14:    not Collinear b,a,c and
A15:    E = Plane(b,a,c) by A11,A2,A3,A4,A5,A6,Th45;
A16:    not Collinear a,b,c by GTARSKI3:45,A14;
        E = Plane(Line(a,b),c) by A15,A14,Def11
         .= Plane(a,b,c) by A16,Def11;
        hence thesis by A16,A11,A13,A2,A3,A4,A5,A6,Th45;
      end;
      suppose not b in A;
        hence thesis by A11,A2,A3,A4,A5,A6,Th45;
      end;
    end;
  end;
