theorem
  a,b '||' c,d iff [[a,b],[c,d]] in lambda(the CONGR of S)
proof
  thus a,b '||' c,d implies [[a,b],[c,d]] in lambda(the CONGR of S)
  proof
    assume 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 thesis by Def1;
  end;
  assume [[a,b],[c,d]] in lambda(the CONGR of S);
  then
  [[a,b],[c,d]] in the CONGR of S or [[a,b],[d,c]] in the CONGR of S by Def1;
  hence a,b // c,d or a,b // d,c;
end;
