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

theorem Prelim01:
  for S being satisfying_CongruenceSymmetry
  satisfying_CongruenceEquivalenceRelation satisfying_CongruenceIdentity
  TarskiGeometryStruct for a,b,c,d being POINT of S st a,b equiv c,d
  holds a,b equiv d,c & b,a equiv c,d & b,a equiv d,c & c,d equiv a,b &
  d,c equiv a,b & c,d equiv b,a & d,c equiv b,a
  proof
    let S be satisfying_CongruenceSymmetry
    satisfying_CongruenceEquivalenceRelation satisfying_CongruenceIdentity
    TarskiGeometryStruct;
    let c1,c2,c3,c4 be POINT of S;
    assume
A1: c1,c2 equiv c3,c4;
    assume
A2: not (c1,c2 equiv c4,c3 & c2,c1 equiv c3,c4 & c2,c1 equiv c4,c3 &
    c3,c4 equiv c1,c2 & c4,c3 equiv c1,c2 & c3,c4 equiv c2,c1 &
    c4,c3 equiv c2,c1);
A3: c1,c2 equiv c2,c1 by GTARSKI1:def 5;
    then
A4: c3,c4 equiv c2,c1 by A1,GTARSKI1:def 6;
    c1,c2 equiv c1,c2 by GTARSKI3:1;
    then
A5: c3,c4 equiv c1,c2 by A1,GTARSKI1:def 6;
    c3,c4 equiv c4,c3 by GTARSKI1:def 5;
    hence contradiction by A1,A2,A3,A4,A5,GTARSKI1:def 6;
  end;
