reserve x for object, X,Y for set;
reserve C for non empty set;
reserve c for Element of C;
reserve f,f1,f2,f3,g,g1 for complex-valued Function;
reserve r,p for Complex;

theorem
  1(#)f = f
proof
A1: now
    let c be object;
    assume c in dom (1(#)f);
    hence (1(#)f).c = 1 * f.c by VALUED_1:def 5
      .= f.c;
  end;
  dom (1(#)f) = dom f by VALUED_1:def 5;
  hence thesis by A1,FUNCT_1:2;
end;
