reserve i,j,k,n,m for Nat,
        X for set,
        b,s for bag of X,
        x for object;
reserve O for Ordinal,
        R for right_zeroed add-associative right_complementable
          right_unital distributive non trivial doubleLoopStr,
        p for Polynomial of O, R;
reserve O for Ordinal,
        R for right_zeroed add-associative right_complementable
             right_unital distributive non trivial doubleLoopStr,
        p for Polynomial of O, R;

theorem Th74:
  for M be Jpolynom of 4,F_Complex
    for x1,x2,x3 be Nat st x1 is square & x2 is square & x3 is square
      ex z be Integer st -z = sqrt x1 + 2* sqrt x2 + 4 *sqrt x3 &
        eval(Jsqrt M,@<%z,x1,4*x2,16*x3%>)=0
proof
  let M be Jpolynom of 4,F_Complex;
  let x1,x2,x3 be Nat such that
A1: x1 is square & x2 is square & x3 is square;
  consider m1 be Nat such that
A2: m1^2 = x1 by A1;
  consider m2 be Nat such that
A3: m2^2 = x2 by A1;
  consider m3 be Nat such that
A4: m3^2 = x3 by A1;
A5: sqrt x1 = m1 & sqrt x2= m2 & sqrt x3= m3 by A3,A4,A2,SQUARE_1:22;
  take z= -( m1 + 2*m2 + 4*m3);
  reconsider A = {} as Subset of Seg 4\{1} by XBOOLE_1:2;
  set f=<%z,x1,4*x2,16*x3%>,c = _sqrt XFS2FS(f);
  set F = <* z,x1,4*x2,16*x3 *>,AF=the addF of F_Complex;
  set S = SignGen(c,AF,A);
A6: XFS2FS f= F by Lm1;
A7:dom S = dom c by HILB10_7:def 11;
  len F = 4 by CARD_1:def 7;
  then
A8:len S = len c = 4 by A6,Def11,CARD_1:def 7;
  then
A9:S/.1=S.1 & S/.2=S.2 & S/.3=S.3 & S/.4=S.4 by A7,PARTFUN1:def 6,FINSEQ_3:25;
  then
A10: S = <*S/.1,S/.2,S/.3,S/.4*> by A8,FINSEQ_4:76;
A11:AF=addcomplex by COMPLFLD:def 1;
A12: AF "**" <*S/.1,S/.2,S/.3,S/.4*> = AF "**" (<*S/.1,S/.2,S/.3*>^<*S/.4*>)
  by FINSEQ_4:74
  .= AF.(AF "**" <*S/.1,S/.2,S/.3*>,S/.4) by FVSUM_1:8,FINSOP_1:4
  .=AF.(AF.(AF.(S/.1,S/.2),S/.3),S/.4) by FINSOP_1:14;
  AF.(S/.1,S/.2) = (S.1) + (S.2) by A11,A9,BINOP_2:def 3;
  then
A13: AF.(AF.(S/.1,S/.2),S/.3) = (S.1) + (S.2) + (S.3) by A11,A9,BINOP_2:def 3;
  A14:@@f= @f = f;
  not 1 in A & 1 in dom S & 2 in dom S & 3 in dom S & 4 in dom S
  by A8,FINSEQ_3:25;
  then
A15: S.1 = c.1 & S.2 = c.2 & S.3 = c.3 & S.4 = c.4 by HILB10_7:def 11;
A16: c.1 = F.1 = z by A6,Def11;
  F.2 = x1;
  then
A17: c.2 = sqrt x1 by Th71,A6;
  F.3 = 4*x2;
  then
A18: c.3 = sqrt (4*x2) by Th71,A6
  .= 2 * sqrt x2 by SQUARE_1:20,29;
  F.4 = 16*x3;
  then
A19: c.4 = sqrt (4*4*x3) by Th71,A6
  .= sqrt (4*4) * sqrt (x3) by SQUARE_1:29
  .= 2* 2 * sqrt x3 by SQUARE_1:20,29;
A20: 4 is non trivial by NEWTON03:def 1;
  AF "**" S = (c.1)+(c.2)+(c.3)+(c.4) by A13,A15,A10,A12,A11,BINOP_2:def 3;
  hence thesis by A5,A14,A20,Th70,A16,A17,A18,A19;
end;
