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 Satz8p10:
  right_angle a,b,c & a,b,c cong a9,b9,c9 implies right_angle a9,b9,c9
  proof
    assume that
A1: right_angle a,b,c and
A2: a,b,c cong a9,b9,c9;
    per cases;
    suppose b = c;
      then a,b equiv a9,b9 & a,b equiv a9,c9 & b,b equiv b9,c9
        by A2,GTARSKI1:def 3;
      then b9 = c9 by GTARSKI1:def 7,GTARSKI3:4;
      hence thesis by Satz8p5;
    end;
    suppose
A3:   b <> c;
      set d  = reflection(b,c),
          d9 = reflection(b9,c9);
A4:   a,b equiv a9,b9 & a,c equiv a9,c9 & b,c equiv b9,c9 by A2,GTARSKI1:def 3;
      now
        Middle c,b,d & Middle c9,b9,d9 by GTARSKI3:def 13;
        hence between c,b,d & between c9,b9,d9 by GTARSKI3:def 12;
        thus b,a equiv b9,a9 & c,b equiv c9,b9 & c,a equiv c9,a9
          by A4,Prelim01;
        b,c equiv reflection(b,b),reflection(b,c) by GTARSKI3:105; then
A5:     b9,c9 equiv reflection(b,b),reflection(b,c) by A4,GTARSKI1:def 6;
        b9,c9 equiv reflection(b9,b9),reflection(b9,c9) by GTARSKI3:105;
        then reflection(b,b),reflection(b,c) equiv
               reflection(b9,b9),reflection(b9,c9) by A5,GTARSKI1:def 6;
        then b,reflection(b,c) equiv reflection(b9,b9),reflection(b9,c9)
          by GTARSKI3:104;
        hence b,d equiv b9,d9 by GTARSKI3:104;
      end;
      then c,b,d,a AFS c9,b9,d9,a9 by GTARSKI3:def 2; then
A6:   a,d equiv d9,a9 by A3,GTARSKI3:6,10;
      a,c equiv d9,a9 by A1,A6,Prelim03;
      hence thesis by A4,Prelim03;
    end;
  end;
