reserve i,j,k for Element of NAT;
reserve a,b for Element of REAL;

theorem Th1:
  REAL c< COMPLEX
proof
  set X = { x where x is Element of Funcs({0,one},REAL): x.one = 0};
  thus REAL c= COMPLEX by XBOOLE_1:7;
A1: now
    assume (0,1) --> (0,1) in X;
    then
    ex x being Element of Funcs({0,one},REAL) st x = (0,1) --> (0,1) & x.
    one = 0;
    hence contradiction by FUNCT_4:63;
  end;
  REAL+ c= REAL+ \/ [:{{}},REAL+:] by XBOOLE_1:7;
  then
A2: REAL+ c= REAL by ARYTM_2:3,ZFMISC_1:34;
  then reconsider z = 0, j = 1 as Element of REAL by ARYTM_2:20;
A3: not (0,1) --> (z,j) in REAL by Lm7;
  rng (0,1) --> (0,1) c= {0,1} & {0,1} c= REAL by A2,ARYTM_2:20,FUNCT_4:62
,ZFMISC_1:32;
  then dom (0,1) --> (0,1) = {0,1} & rng (0,1) --> (0,1) c= REAL by FUNCT_4:62;
  then (0,1) --> (0,1) in Funcs({0,one},REAL) by FUNCT_2:def 2;
  then (0,1) --> (0,1) in Funcs({0,one},REAL) \ X by A1,XBOOLE_0:def 5;
  hence thesis by A3,XBOOLE_0:def 3;
end;
