reserve            S for satisfying_CongruenceSymmetry
                         satisfying_CongruenceEquivalenceRelation
                         TarskiGeometryStruct,
         a,b,c,d,e,f for POINT of S;

theorem EQUIV1:
  S is satisfying_SAS iff S is satisfying_SST_A5
  proof
    thus S is satisfying_SAS implies S is satisfying_SST_A5
    proof
      assume
A1:   S is satisfying_SAS;
      now
        let a,b,c,d,a9,b9,c9,d9 be POINT of S;
        assume
A2:     a <> b & between a,b,c & between a9,b9,c9 & a,b equiv a9,b9 &
          b,c equiv b9,c9 & a,d equiv a9,d9 & b,d equiv b9,d9;
        then a,b,d cong a9,b9,d9;
        then d,c equiv d9,c9 by A1,A2;
        then c,d equiv d9,c9 by Satz2p4;
        hence c,d equiv c9,d9 by Satz2p5;
      end;
      hence thesis;
    end;
    thus S is satisfying_SST_A5 implies S is satisfying_SAS
    proof
      assume
A3:   S is satisfying_SST_A5;
      now
        let a, b, c, x, a1, b1, c1, x1 be POINT of S;
        assume a <> b & a,b,c cong a1,b1,c1 & between a,b,x &
          between a1,b1,x1 & b,x equiv b1,x1;
        then x,c equiv x1,c1 by A3;
        then c,x equiv x1,c1 by Satz2p4;
        hence c,x equiv c1,x1 by Satz2p5;
      end;
      hence thesis;
    end;
  end;
