reserve p,q for Rational;
reserve g,m,m1,m2,n,n1,n2 for Nat;
reserve i,i1,i2,j,j1,j2 for Integer;
reserve R for Ring, F for Field;

theorem Th96:
for F being 0-characteristic Field,
    i,j being non zero Integer st j divides i
holds (i div j) '*' 1.F = (i '*' 1.F) * (j '*' 1.F)"
proof
let F be 0-characteristic Field, i,j be non zero Integer;
A1: Char F = 0 by Def6;
assume j divides i;
then consider k being Integer such that A2: i = j * k;
A3: (i div j) * j = [\  k * (j / j) /] * j by A2
                 .= [\ k*1 /] * j by XCMPLX_1:60
                 .= i by A2;
A4: j '*' 1.F <> 0.F
 proof
 per cases;
 suppose j > 0;
   then j is Element of NAT by INT_1:3;
   hence thesis by A1,Def5;
  end;
 suppose j < 0;
  then A5: - j is Element of NAT by INT_1:3;
  A6: j '*' 1.F = (-(-j)) '*' 1.F .= -((-j) '*' 1.F) by Th62;
  now assume j '*' 1.F = 0.F;
    then -(-((-j) '*' 1.F)) = 0.F by A6;
    hence (-j) '*' 1.F = 0.F;
    end;
  hence thesis by A5,A1,Def5;
  end;
 end;
A7: i '*' 1.F <> 0.F
 proof
 per cases;
 suppose i > 0;
   then i is Element of NAT by INT_1:3;
   hence thesis by A1,Def5;
  end;
 suppose i < 0;
  then A8: - i is Element of NAT by INT_1:3;
  A9: i '*' 1.F = (-(-i)) '*' 1.F .= -((-i) '*' 1.F) by Th62;
  now assume i '*' 1.F = 0.F;
    then -(-((-i) '*' 1.F)) = 0.F by A9;
    hence (-i) '*' 1.F = 0.F;
    end;
  hence thesis by A8,A1,Def5;
  end;
 end;
(((i div j) '*' 1.F) *  (i '*' 1.F)") * (j '*' 1.F)
   = (i '*' 1.F)" *  (((i div j) '*' 1.F) * (j '*' 1.F)) by GROUP_1:def 3
  .= (i '*' 1.F)" *  ((i div j)*j '*' 1.F) by Th66
  .= 1.F by A3,A7,VECTSP_1:def 10;
then (j '*' 1.F)"
       = ((i div j) '*' 1.F) *  (i '*' 1.F)" by A4,VECTSP_1:def 10;
hence (i '*' 1.F) * (j '*' 1.F)"
       = ((i '*' 1.F) *  (i '*' 1.F)") * ((i div j) '*' 1.F) by GROUP_1:def 3
      .= 1.F * ((i div j) '*' 1.F) by A7,VECTSP_1:def 10
      .= ((i div j) '*' 1.F);
end;
