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 Th9:
  r <> s & s,c <= r,a & between a,A,c & r in A & are_orthogonal A,r,a &
  s in A & are_orthogonal A,s,c implies ((Middle r,m,s implies
  (for u be POINT of S holds (r out u,a iff s out reflection(m,u),c))) &
  (for u,v be POINT of S st r out u,a & s out v,c holds between u,A,v))
  proof
    assume that
A1: r <> s and
A2: s,c <= r,a and
A3: between a,A,c and
A4: r in A and
A5: are_orthogonal A,r,a and
A6: s in A and
A7: are_orthogonal A,s,c;
    consider t be POINT of S such that
A8: t in A and
A9: between a,t,c by A3;
A10: a <> r & are_orthogonal A,r,Line(r,a) by A5,GTARSKI4:def 4;
A11: c <> s & are_orthogonal A,s,Line(s,c) by A7,GTARSKI4:def 4;
    consider b be POINT of S such that
A12: between r,b,a and
A13: s,c equiv r,b by A2;
        G1: Line(s,r) is_line & Line(c,s) is_line by A1,A7,GTARSKI4:def 4;
        G2: s in Line(c,s) & s in Line(s,r) by GTARSKI3:83;
      now    let u9,v9 be POINT of S;
          assume
      u9 in Line(c,s) &
      v9 in Line(s,r);
          then v9 in A & u9 in Line(s,c) by A3,A4,A6,A1,GTARSKI3:87;
          hence right_angle u9,s,v9 by A11,GTARSKI4:13;
      end;
      then R1: are_orthogonal c,s,s,r
by G1,G2,A7,GTARSKI4:def 2,def 3,def 4,A1;
        G1: Line(a,r) is_line & Line(r,s) is_line by A1,A5,GTARSKI4:def 4;
        G2: r in Line(a,r) & r in Line(r,s) by GTARSKI3:83;
      now
          let u9,v9 be POINT of S;
          assume u9 in Line(a,r) & v9 in Line(r,s);
         then v9 in A & u9 in Line(r,a) by A4,A6,A3,A1,GTARSKI3:87;
          hence right_angle u9,r,v9 by A10,GTARSKI4:13;
      end;
      then are_orthogonal Line(a,r),Line(s,r) by G1,G2,GTARSKI4:def 2;
      then R2: are_orthogonal a,r,s,r by A5,A1,GTARSKI4:def 4;
      A = Line(r,s) by A4,A6,A3,A1,GTARSKI3:87;
      then ex x9 be POINT of S st t = x9 & Collinear r,s,x9 by A8;
      then Collinear s,r,t & between c,t,a by A9,GTARSKI3:14;
    then consider m be POINT of S such that
A18: Middle s,m,r and
A19: Middle c,m,b by R1,R2,A12,A13,GTARSKI4:40;
    now
      hereby
        let m9 be POINT of S;
        assume
A20:    Middle r,m9,s;
        then
A21:    Middle s,m9,r by GTARSKI3:96;
        then
A22:    m9 = m by A18,GTARSKI3:108;
        let u be POINT of S;
        hereby
          assume
A23:      r out u,a;
            G1: s <> reflection(m,u)
            proof
              assume s = reflection(m,u);
              then Middle s,m,u by GTARSKI3:96,def 13;
              hence contradiction by A18,GTARSKI3:99,A23;
            end;
            (between r,u,b or between r,b,u) & s = reflection(m,r) &
              c = reflection(m,b)
              by A18,GTARSKI3:18,58,96,def 13,A23,A12,A19;
            then between s,reflection(m,u),c or between s,c,reflection(m,u)
              by GTARSKI3:106;
            hence s out reflection(m9,u),c
              by A21,A18,G1,A7,GTARSKI4:def 4,GTARSKI3:108;
        end;
        assume
A24:    s out reflection(m9,u),c;
A25:    s = reflection(m9,reflection(m9,s)) &
          c = reflection(m9,reflection(m9,c)) by GTARSKI3:101;
        r = reflection(m,s) & b = reflection(m,c)
          by A18,A19,GTARSKI3:def 13;
        then
A26:    between r,u,b or between r,b,u by A25,A22,A24,GTARSKI3:106;
        r <> b by A13,A7,GTARSKI4:def 4,GTARSKI1:def 7;
        hence r out u,a
          by A26,A12,GTARSKI3:56,18,A24,A20,Th1,A2,GTARSKI3:def 13;
      end;
      hereby
        let u,v be POINT of S;
        assume that
A27:    r out u,a and
A28:    s out v,c;
        set u9 = reflection(m,u);
          G1: s <> u9
          proof
            assume s = u9;
            then Middle s,m,u by GTARSKI3:def 13,96;
            hence contradiction by A27,A18,GTARSKI3:99;
          end;
A30:      r = reflection(m,s) & b = reflection(m,c)
            by A18,A19,GTARSKI3:def 13;
          between r,u,b or between r,b,u by A27,A12,GTARSKI3:18,58;
          then between reflection(m,s),
                       reflection(m,reflection(m,u)),
                       reflection(m,c) or
               between reflection(m,s),
                       reflection(m,c),
                       reflection(m,reflection(m,u)) by A30,GTARSKI3:101;
         then G3: between s,u9,c or between s,c,u9 by GTARSKI3:106;
          Collinear r,s,m by A18;
          then m in Line(r,s);
          then A31:m in A by A1,A3,A4,A6,GTARSKI3:87;
A32:        not u in A
            proof
              assume u in A;
              then
A33:          Line(r,u) = A by A27,A4,A3,GTARSKI3:87;
              Collinear r,u,a by A27,GTARSKI3:14;
              hence contradiction by A33,A3;
            end;
            GG2: not u9 in A
            proof
              assume
A34:          u9 in A;
              Middle u,m,u9 by GTARSKI3:def 13;
              then Collinear u9,m,u by GTARSKI3:14;
              then
A35:          u in Line(u9,m);
              u9 <> m
              proof
                assume u9 = m;
                then Middle u,m,m by GTARSKI3:def 13;
                hence contradiction by A32,A31,GTARSKI1:def 7;
              end;
              hence contradiction by A35,A32,A34,A31,A3,GTARSKI3:87;
            end;
            Middle u9,m,u by GTARSKI3:96,def 13;
          then U2: between u9,A,u by A31,A32,A3,GG2;
          U3: Middle u9,m,u by GTARSKI3:96,def 13;
           s in A & s out u9,v by A6,G1,G3,A28,GTARSKI3:18,56,58;
        then between v,A,u by U2,U3,A31,Th7;
        hence between u,A,v by GTARSKI3:14;
      end;
    end;
    hence thesis;
  end;
