reserve a,b,c,k,m,n for Nat;
reserve i,j,x,y for Integer;
reserve p,q for Prime;
reserve r,s for Real;

theorem Th62:
  p = 4*k+3 & p divides i^2+j^2 implies p divides i & p divides j
  proof
    assume
A1: p = 4*k+3 & p divides i^2+j^2;
A2: i^2 = (-i)^2;
A3: j^2 = (-j)^2;
    per cases;
    suppose i >= 0 & j >= 0;
      then i in NAT & j in NAT by INT_1:3;
      hence thesis by A1,Lm17;
    end;
    suppose i >= 0 & j < 0;
      then i in NAT & -j in NAT by INT_1:3;
      then p divides i & p divides -j by A1,A3,Lm17;
      hence thesis by INT_2:10;
    end;
    suppose i < 0 & j >= 0;
      then -i in NAT & j in NAT by INT_1:3;
      then p divides -i & p divides j by A1,A2,Lm17;
      hence thesis by INT_2:10;
    end;
    suppose i < 0 & j < 0;
      then -i in NAT & -j in NAT by INT_1:3;
      then p divides -i & p divides -j by A1,A2,A3,Lm17;
      hence thesis by INT_2:10;
    end;
  end;
