
theorem Th11:
  for n,m being Integer,n2,m2 being Element of NAT holds (m=0 & n2
=n & m2=m implies n div m=0 & n2 div m2=0)& (n>=0 & m>0 & n2=n & m2=m implies n
  div m =n2 div m2)& (n>=0 & m<0 & n2=n & m2= -m implies (m2*(n2 div m2)=n2
  implies n div m =-(n2 div m2)) &(m2*(n2 div m2)<>n2 implies n div m =-(n2 div
m2)-1))& (n<0 & m>0 & n2= -n & m2=m implies (m2*(n2 div m2)=n2 implies n div m
=-(n2 div m2)) &(m2*(n2 div m2)<>n2 implies n div m =-(n2 div m2)-1))& (n<0 & m
  <0 & n2= -n & m2= -m implies n div m =n2 div m2)
proof
  let n,m be Integer,n2,m2 be Element of NAT;
  thus m=0 & n2=n & m2=m implies n div m=0 & n2 div m2=0;
  thus n>=0 & m>0 & n2=n & m2=m implies n div m =n2 div m2;
  thus n>=0 & m<0 & n2=n & m2= -m implies (m2*(n2 div m2)=n2 implies n div m =
  -(n2 div m2)) &(m2*(n2 div m2)<>n2 implies n div m =-(n2 div m2)-1)
  proof
    assume that
    n>=0 and
A1: m<0 and
A2: n2=n and
A3: m2= -m;
    thus m2*(n2 div m2)=n2 implies n div m =-(n2 div m2)
    proof
      assume
A4:   m2*(n2 div m2)=n2;
   m2>0 by A1,A3,XREAL_1:58;
      then
A6:   n2 = m2 * (n2 div m2) + (n2 mod m2) by NAT_D:2;
      thus n div m =[\ n/m /] by INT_1:def 9
        .=[\ (-n)/(-m) /] by XCMPLX_1:191
        .=(-n2) div m2 by A2,A3,INT_1:def 9
        .= -(n2 div m2) by A4,A6,WSIERP_1:42;
    end;
    thus m2*(n2 div m2)<>n2 implies n div m =-(n2 div m2)-1
    proof
      assume
A7:   m2*(n2 div m2)<>n2;
A8:   m2>0 by A1,A3,XREAL_1:58;
      then n2 = m2 * (n2 div m2) + (n2 mod m2) by NAT_D:2;
      then not n2 mod m2=0 by A7;
      then
A9:   (-n2) div m2 +1= -(n2 div m2) by A8,WSIERP_1:41;
      n div m =[\ n/m /] by INT_1:def 9
        .=[\ (-n)/(-m) /] by XCMPLX_1:191
        .=(-n2) div m2 by A2,A3,INT_1:def 9;
      hence thesis by A9;
    end;
  end;
  thus n<0 & m>0 & n2= -n & m2=m implies (m2*(n2 div m2)=n2 implies n div m =-
  (n2 div m2)) &(m2*(n2 div m2)<>n2 implies n div m =-(n2 div m2)-1)
  proof
    assume that
    n<0 and
A10: m>0 and
A11: n2= -n and
A12: m2=m;
    thus m2*(n2 div m2)=n2 implies n div m =-(n2 div m2)
    proof
A13:  n2 = m2 * (n2 div m2) + (n2 mod m2) by A10,A12,NAT_D:2;
      assume
A14:  m2*(n2 div m2)=n2;
      (n div m)=(-n2) div m by A11
        .=-(n2 div m2) by A12,A14,A13,WSIERP_1:42;
      hence thesis;
    end;
    thus m2*(n2 div m2)<>n2 implies n div m =-(n2 div m2)-1
    proof
      assume
A15:  m2*(n2 div m2)<>n2;
      n2 = m2 * (n2 div m2) + (n2 mod m2) by A10,A12,NAT_D:2;
      then not n2 mod m2=0 by A15;
      then (-n2) div m2 +1 = -(n2 div m2) by A10,A12,WSIERP_1:41;
      hence thesis by A11,A12;
    end;
  end;
  thus n<0 & m<0 & n2= -n & m2= -m implies n div m =n2 div m2
  proof
    assume that
    n<0 and
    m<0 and
A16: n2= -n & m2= -m;
    thus n div m = [\ n/m /] by INT_1:def 9
      .=[\ (-n)/(-m) /] by XCMPLX_1:191
      .=n2 div m2 by A16,INT_1:def 9;
  end;
end;
