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 Satz2p12: ::GTARSKI1:26
  q <> a implies (for x1,x2 st between q,a,x1 & a,x1 equiv b,c &
  between q,a,x2 & a,x2 equiv b,c holds x1 = x2)
  proof
    assume
A1: q <> a;
A2: S is satisfying_SST_A5;
    hereby
      let x1,x2;
      assume
A3:   between q,a,x1 & a,x1 equiv b,c & between q,a,x2 & a,x2 equiv b,c;
      then b,c equiv a,x2 by Satz2p2; then
A4:   a,x1 equiv a,x2 by A3,Satz2p3;
      q,a equiv q,a & a,x1 equiv a,x1 by Satz2p1;
      then q,a,x1,x1 AFS q,a,x1,x2 by A3,A4,Satz2p11;
      then x1,x1 equiv x1,x2 by A1,A2;
      hence x1 = x2 by Satz2p2,GTARSKI1:def 7;
    end;
  end;
