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 Th49:
  not Collinear a,b,c implies ex E being Subset of S st Plane(a,b,c) = E &
  E is_plane & a in E & b in E & c in E
  proof
    assume
A1: not Collinear a,b,c;
    set E = Plane(a,b,c);
A2: E = Plane(Line(a,b),c) by A1,Def11;
    take E;
    a <> b by A1,GTARSKI3:46;
    then
A3: Line(a,b) is_line;
A4: not c in Line(a,b)
    proof
      assume c in Line(a,b);
      then ex x be POINT of S st c = x & Collinear a,b,x;
      hence contradiction by A1;
    end;
    Line(a,b) c= E by A3,A4,A2,Th31;
    hence thesis by A1,A2,A3,Th48,GTARSKI3:83;
  end;
