reserve            S for satisfying_CongruenceSymmetry
                         satisfying_CongruenceEquivalenceRelation
                         TarskiGeometryStruct,
         a,b,c,d,e,f for POINT of S;
reserve S for satisfying_CongruenceSymmetry
              satisfying_CongruenceEquivalenceRelation
              satisfying_CongruenceIdentity
              satisfying_SegmentConstruction
              satisfying_SAS
              TarskiGeometryStruct,
        q,a,b,c,a9,b9,c9,x1,x2 for POINT of S;

theorem Satz2p11: ::GTARSKI1:24
  between a,b,c & between a9,b9,c9 & a,b equiv a9,b9 & b,c equiv b9,c9
  implies a,c equiv a9,c9
  proof
    assume
A1: between a,b,c & between a9,b9,c9 & a,b equiv a9,b9 & b,c equiv b9,c9;
A2: S is satisfying_SST_A5;
    b,a equiv a9,b9 by A1,Satz2p4; then
A3: a,b,c,a AFS a9,b9,c9,a9 by A1,Satz2p5,Satz2p8;
    per cases;
    suppose a = b;
      hence thesis by A1,Satz2p2,GTARSKI1:def 7;
    end;
    suppose a <> b;
      then c,a equiv c9,a9 by A3,A2;
      then a,c equiv c9,a9 by Satz2p4;
      hence thesis by Satz2p5;
    end;
  end;
