reserve a,b,k,m,n,s for Nat;
reserve c,c1,c2,c3 for Complex;
reserve i,j,z for Integer;
reserve p for Prime;
reserve x for object;
reserve f,g for complex-valued FinSequence;

theorem Th58:
  for f being complex-valued Function holds (c(#)f)" = c" (#) f"
  proof
    let f be complex-valued Function;
A1: dom((c(#)f)") = dom(c(#)f) by VALUED_1:def 7;
A2: dom(c(#)f) = dom f by VALUED_1:def 5;
    dom(f") = dom(f) by VALUED_1:def 7;
    hence dom((c(#)f)") = dom(c"(#)f") by A1,A2,VALUED_1:def 5;
    let x;
    assume x in dom((c(#)f)");
A3: (c(#)f).x = c*f.x by VALUED_1:6;
A4: f".x = (f.x)" by VALUED_1:10;
    thus ((c(#)f)").x = ((c(#)f).x)" by VALUED_1:10
    .= c"*(f.x)" by A3,XCMPLX_1:204
    .= (c"(#)f").x by A4,VALUED_1:6;
  end;
