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 Th76:
  A is_plane & not a in A implies ex c st between2 a,A,c
  proof
    assume that
A1: A is_plane and
A2: not a in A;
    consider p,q,r such that
A3: not Collinear p,q,r and
A4: A = Plane(p,q,r) by A1;
A5: not r in Line(p,q)
    proof
      assume r in Line(p,q);
      then ex x be POINT of S st r = x & Collinear p,q,x;
      hence thesis by A3;
    end;
    p <> q by A3,GTARSKI3:46;
    then
A6: A = Plane(Line(p,q),r) & Line(p,q) is_line by A3,Def11,A4;
    then Line(p,q) c= A by A5,Th31;
    then
A7: p in A & q in A & r in A by A5,Th31,A6,GTARSKI3:83;
    set c = reflection(p,a);
A8: p <> c
    proof
      assume p = c;
      then Middle a,c,c by GTARSKI3:def 13;
      hence contradiction by A2,A7,GTARSKI3:104,GTARSKI1:def 7;
    end;
    take c;
A9:   Middle a,p,c by GTARSKI3:def 13;
      Collinear p,c,a by A9;
      then
A10:   a in Line(p,c);
      not c in A
      proof
        assume c in A;
        then Line(p,c) c= A by A7,A3,A8,A4,Th69;
        hence thesis by A10,A2;
      end;
    hence thesis by A9,A1,A2,A7;
  end;
