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 Prelim03:
  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 or
  a,b equiv p,q or b,a equiv p,q or a,b equiv q,p or b,a equiv q,p) &
  (p,q equiv c,d or p,q equiv d,c or q,p equiv c,d or q,p equiv d,c or
  c,d equiv p,q or d,c equiv p,q or c,d equiv q,p or d,c equiv q,p)
  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 & a,b equiv c,d
  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 or
      a,b equiv p,q or b,a equiv p,q or a,b equiv q,p or b,a equiv q,p) &
      (p,q equiv c,d or p,q equiv d,c or q,p equiv c,d or q,p equiv d,c or
      c,d equiv p,q or d,c equiv p,q or c,d equiv q,p or d,c equiv q,p);
    then p,q equiv a,b & p,q equiv c,d by Prelim01;
    hence thesis by Prelim02;
  end;
