reserve S for non empty satisfying_Tarski-model
              TarskiGeometryStruct,
        a,b,c,d,c9,x,y,z,p,q,q9 for POINT of S;
reserve              S for satisfying_Tarski-model TarskiGeometryStruct,
        a,a9,b,b9,c,c9 for POINT of S;

theorem Satz8p7:
  right_angle a,b,c & right_angle a,c,b implies b = c
  proof
    assume that
A1: right_angle a,b,c and
A2: right_angle a,c,b;
    assume
A3: b <> c;
    set c9 = reflection(b,c),
        a9 = reflection(c,a);
    now
      a9,c equiv a9,c9
      proof
        now
          now
            thus a,c9 equiv a,c by A1,GTARSKI3:4;
            right_angle c,c,a by Satz8p2,Satz8p5;
            hence a,c equiv a9,c by Prelim01;
          end;
          hence a,c9 equiv a9,c by GTARSKI3:5;
          now
            thus right_angle b,c,a by A2,Satz8p2;
            thus b <> c by A3;
            Middle c,b,c9 by GTARSKI3:def 13;
            hence Collinear c,b,c9 by Prelim08a;
          end;
          then right_angle c9,c,a by Satz8p3; then
A4:       right_angle a,c,c9 by Satz8p2;
          a,reflection(c,c9) equiv
            reflection(c,a),reflection(c,reflection(c,c9)) by GTARSKI3:105;
          then a,c9 equiv a9,reflection(c,reflection(c,c9)) by A4,GTARSKI3:5;
          hence a,c9 equiv a9,c9 by GTARSKI3:101;
        end;
        hence thesis by GTARSKI1:def 6;
      end;
      hence right_angle a9,b,c;
      Middle a,c,a9 by GTARSKI3:def 13;
      hence between a,c,a9 by GTARSKI3:def 12;
    end;
    hence contradiction by A1,A3,Satz8p6;
  end;
