theorem Th31:
  for Y being complex-functions-membered set, f being PartFunc of X,Y holds
  f[#]c = f<#>(dom f-->c)
  proof
    let Y be complex-functions-membered set, f be PartFunc of X,Y;
    set g = dom f-->c;
A1: dom (f[#]c) = dom f by VALUED_2:def 39;
A2: dom (f<#>g) = dom f /\ dom g by VALUED_2:def 43;
    now
      let x be object;
      assume
A4:   x in dom (f[#]c);
      hence (f[#]c).x = c(#)(f.x) by VALUED_2:def 39
      .= (f.x)(#)(g.x) by A1,A4,FUNCOP_1:7
      .= (f<#>g).x by A1,A2,A4,VALUED_2:def 43;
    end;
    hence thesis by A1,A2;
  end;
