 reserve a,b,r for Complex;
 reserve V for ComplexLinearSpace;
reserve A,B for non empty set;
reserve f,g,h for Element of PFuncs(A,COMPLEX);

theorem Th11:
  (multcomplexcpfunc A).(1r,f) = f
proof
  reconsider g = (multcomplexcpfunc A).(1r,f)
  as Element of PFuncs(A,COMPLEX);
A1: now
    let x be Element of A;
    assume x in dom f;
    hence g.x = 1r*(f.x) by Th7
      .= f.x by COMPLEX1:def 4;
  end;
  dom g=dom f by Th7;
  hence thesis by A1,PARTFUN1:5;
end;
