theorem Th71:
 for R being non empty Relation, x being object
  holds Im(R,x) = { I`2 where I is Element of R: I`1 = x }
 proof let R be non empty Relation, x being object;
   set X = { I`2 where I is Element of R: I`1 = x };
  thus Im(R,x) c= X
   proof let z be object;
    assume z in Im(R,x);
     then consider y being object such that
A1:   [y,z] in R and
A2:   y in {x} by RELAT_1:def 13;
A3:   y = x by A2,TARSKI:def 1;
     y = [y,z]`1 & z = [y,z]`2;
    hence z in X by A1,A3;
   end;
  let z be object;
  assume z in X;
   then consider I being Element of R such that
A4: z= I`2 and
A5: I`1 = x;
A6: I = [I`1,I`2] by Th15;
   x in {x} by TARSKI:def 1;
  hence z in Im(R,x) by A4,A5,A6,RELAT_1:def 13;
 end;
