reserve a,b,c,d,x,y,w,z,x1,x2,x3,x4 , X for set;
reserve A for non empty set;

theorem Th2:
  for g being Quaternion
  ex r,s,t,u being Real st g = [*r,s,t,u*]
proof
  let g be Quaternion;
A1:g in QUATERNION by Def2;
  per cases;
  suppose g in COMPLEX;
    then consider r,s being Element of REAL such that
A2: g = [*r,s*] by ARYTM_0:9;
    take r,s,0,0;
    thus thesis by A2,Def5,A1;
  end;
  suppose not g in COMPLEX;
    then
A3: g in Funcs(4,REAL) \ { x where x is Element of Funcs(4,REAL):
    x.2=0 & x.3=0} by A1,XBOOLE_0:def 3;
    then consider f being Function such that
A4: g = f and
A5: dom f = 4 and
A6: rng f c= REAL by FUNCT_2:def 2;
A7: 0 in 4 by CARD_1:52,ENUMSET1:def 2;
A8: 1 in 4 by CARD_1:52,ENUMSET1:def 2;
A9: 2 in 4 by CARD_1:52,ENUMSET1:def 2;
A10: 3 in 4 by CARD_1:52,ENUMSET1:def 2;
A11: f.0 in rng f by A5,A7,FUNCT_1:3;
A12: f.1 in rng f by A5,A8,FUNCT_1:3;
A13: f.2 in rng f by A5,A9,FUNCT_1:3;
    f.3 in rng f by A5,A10,FUNCT_1:3;
    then reconsider
    r = f.0, s = f.1, t=f.2, u=f.3 as Element of REAL by A6,A11,A12,A13;
A14: g = (0,1,2,3)-->(r,s,t,u) by A4,A5,FUNCT_4:144,CARD_1:52;
    take r,s,t,u;
    now
      assume that
A15:  t=0 and
A16:  u=0;
A17:  (0,1,2,3)-->(r,s,t,u).2 = 0 by A15,FUNCT_4:140;
      (0,1,2,3)-->(r,s,t,u).3 = 0 by A16,FUNCT_4:139;
      then g in { x where x is Element of Funcs(4,REAL):
      x.2=0 & x.3=0} by A3,A14,A17;
      hence contradiction by A3,XBOOLE_0:def 5;
    end;
    hence thesis by A14,Def5;
  end;
end;
