reserve a,a1,a2,b,c,d for Ordinal,
  n,m,k for Nat,
  x,y,z,t,X,Y,Z for set;

theorem Th5:
  for f being Function, r being Relation
  for x being object holds x in f.:r iff
  ex y,z being object st [y,z] in r & [y,z] in dom f & f.(y,z) = x
  proof
    let f be Function;
    let r be Relation;
    let x be object;
    hereby assume x in f.:r; then
      consider t being object such that
A1:   t in dom f & t in r & x = f.t by FUNCT_1:def 6;
      consider y,z being object such that
A2:   t = [y,z] by A1,RELAT_1:def 1;
      f.(y,z) = f.[y,z];
      hence ex y,z being object st [y,z] in r & [y,z] in dom f & f.(y,z) = x
              by A1,A2;
    end;
    thus thesis by FUNCT_1:def 6;
  end;
