reserve x,y for set;
reserve X for non empty set;
reserve a,b,c,d for Element of X;
reserve S for OAffinSpace;
reserve a,b,c,d,p,q,r,x,y,z,t,u,w for Element of S;
reserve AS for non empty AffinStruct;

theorem Th38:
  AS=Lambda(S) implies for a,b,c,d being Element of S, a9,b9,c9,d9
being Element of AS st a=a9 & b=b9 & c =c9 & d=d9 holds a9,b9 // c9,d9 iff a,b
  '||' c,d
proof
  assume
A1: AS=Lambda(S);
  let a,b,c,d be Element of S;
  let a9,b9,c9,d9 be Element of AS;
  assume
A2: a=a9 & b=b9 & c =c9 & d=d9;
  thus a9,b9 // c9,d9 implies a,b '||' c,d
  proof
    assume
A3: [[a9,b9],[c9,d9]] in the CONGR of AS;
    assume not [[a,b],[c,d]] in the CONGR of S;
    hence [[a,b],[d,c]] in the CONGR of S by A1,A2,A3,Def1;
  end;
  assume a,b '||' c,d;
  then a,b // c,d or a,b // d,c;
  then [[a,b],[c,d]] in the CONGR of S or [[a,b],[d,c]] in the CONGR of S;
  hence [[a9,b9],[c9,d9]] in the CONGR of AS by A1,A2,Def1;
end;
