reserve a,b,c,d,x,y,w,z,x1,x2,x3,x4 , X for set;
reserve A for non empty set;
reserve i,j,k for Element of NAT;
reserve a,b,c,d for Real;
reserve y,r,s,x,t,w for Element of RAT+;

theorem Th4:
  not (0,1,2,3) --> (a,b,c,d) in COMPLEX
proof
  set f = (0,1,2,3)-->(a,b,c,d);
A1: not f in REAL by Lm5;
  not f in Funcs({0,1},REAL)
  proof
    assume f in Funcs({0,1},REAL);
    then dom f = {0,1} by FUNCT_2:92;
    then {0,1,2,3} c= {0,1} by FUNCT_4:137;
    then {0,1} \/ {2,3} c= {0,1} by ENUMSET1:5;
    hence contradiction by XBOOLE_1:11,ZFMISC_1:22;
  end;
  then not f in Funcs({0,1},REAL)\
  { x where x is Element of Funcs({0,1},REAL): x.1 = 0};
  hence thesis by A1,NUMBERS:def 2,XBOOLE_0:def 3;
