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 Th50:
  A is_line & not c in A implies ex E being Subset of S st (E is_plane &
  A c= E & c in E) & Plane(A,c) = E
  proof
    assume that
A1: A is_line and
A2: not c in A;
    consider a,b be POINT of S such that a <> b and
A3: A = Line(a,b) by A1;
    set E = Plane(A,c);
A4:   not Collinear a,b,c by A2,A3;
      then E = Plane(a,b,c) by A3,Def11;
      then U1: E is_plane by A4;
      U2: A c= E by A1,A2,Th31;
      c in Plane(A,c) by A1,A2,Th48;
    hence thesis by U1,U2;
  end;
