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+;
reserve z,z1,z2,z3,z4 for Quaternion;

theorem Th17:
  z = [*Rea z, Im1 z, Im2 z, Im3 z*]
proof
A1: z in QUATERNION by Def2;
  per cases;
  suppose
A2: z in COMPLEX;
    then
A3: Im2 z = 0 by Def14;
A4: Im3 z = 0 by A2,Def15;
A5: ex z9 being Complex st ( z = z9)&( Rea z = Re z9) by A2,Def12;
    consider z9 being Complex such that
A6: z = z9 and
A7: Im1 z = Im z9 by A2,Def13;
    reconsider Rz = Rea z, Imz = Im1 z as Element of REAL by XREAL_0:def 1;
    [*Rz, Imz*] = z9 by A5,A6,A7,Lm8;
    hence thesis by A3,A4,A6,Lm3;
  end;
  suppose
A8: not z in COMPLEX;
    then
A9: ex f being Function of 4,REAL st ( z = f)&( Im1 z = f.1) by Def13;
A10: ex f being Function of 4,REAL st ( z = f)&( Rea z = f.0) by A8,Def12;
A11: ex f being Function of 4,REAL st ( z = f)&( Im2 z = f.2) by A8,Def14;
A12: ex f being Function of 4,REAL st ( z = f)&( Im3 z = f.3) by A8,Def15;
    consider a,b,c,d being Real such that
A13: z = (0,1,2,3)-->(a,b,c,d) by A9,Th15;
    reconsider a,b,c,d as Element of REAL by XREAL_0:def 1;
A14: z = (0,1,2,3)-->(a,b,c,d) by A13;
A15: Rea z = a by A10,A14,FUNCT_4:142;
A16: Im1 z = b by A9,A14,FUNCT_4:141;
A17: Im2 z = c by A11,A14,FUNCT_4:140;
A18: Im3 z = d by A12,A14,FUNCT_4:139;
    z in Funcs(4,REAL) \ { x where x is Element of Funcs(4,REAL):
    x.2 = 0 & x.3 = 0} by A1,A8,XBOOLE_0:def 3;
    then
A19: not z in { x where x is Element of Funcs(4,REAL):
    x.2 = 0 & x.3 = 0} by XBOOLE_0:def 5;
    reconsider f = z as Element of Funcs(4,REAL) by A14,CARD_1:52,FUNCT_2:8;
    f.2 <> 0 or f.3 <> 0 by A19;
    then c <> 0 or d <> 0 by A14,FUNCT_4:139,140;
    hence thesis by A14,A15,A16,A17,A18,Def5;
  end;
end;
