theorem Th43:
  [x,y] in R*id Y iff y in Y & [x,y] in R
proof
  thus [x,y] in R*id Y implies y in Y & [x,y] in R
  proof
    assume [x,y] in R*id Y;
    then consider z such that
A1: [x,z] in R and
A2: [z,y] in id Y by Def6;
    z = y by A2,Def8;
    hence thesis by A1,A2,Def8;
  end;
  y in Y implies [y,y] in id Y by Def8;
  hence thesis by Def6;
end;
