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 Prelim02:
  for S being satisfying_CongruenceSymmetry
  satisfying_CongruenceEquivalenceRelation satisfying_CongruenceIdentity
  TarskiGeometryStruct for p,q,a,b,c,d being POINT of S st
  (p,q equiv a,b or p,q equiv b,a or q,p equiv a,b or q,p equiv b,a) &
  (p,q equiv c,d or p,q equiv d,c or q,p equiv c,d or q,p equiv d,c)
  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 p,q,a,b,c,d be POINT of S;
    assume (p,q equiv a,b or p,q equiv b,a or q,p equiv a,b or q,p equiv b,a) &
      (p,q equiv c,d or p,q equiv d,c or q,p equiv c,d or q,p equiv d,c);
    then p,q equiv a,b & p,q equiv c,d by Prelim01;
    then a,b equiv c,d by GTARSKI1:def 6;
    hence thesis by Prelim01;
  end;
