reserve i,i1,i2,i3,i4,i5,j,r,a,b,x,y for Integer,
  d,e,k,n for Nat,
  fp,fk for FinSequence of INT,
  f,f1,f2 for FinSequence of REAL,
  p for Prime;
reserve fr for FinSequence of REAL;

theorem Th7:
  p divides i*j implies p divides i or p divides j
proof
  assume
A1: p divides i*j;
  per cases;
  suppose
    i>=0 & j>=0;
    then reconsider i,j as Element of NAT by INT_1:3;
    p divides i*j by A1;
    hence thesis by NEWTON:80;
  end;
  suppose
    i>=0 & j<0;
    then reconsider i,j9=-j as Element of NAT by INT_1:3;
    p divides -(i*j) by A1,INT_2:10;
    then p divides i*j9;
    then p divides i or p divides j9 by NEWTON:80;
    hence thesis by INT_2:10;
  end;
  suppose
    i<0 & j>=0;
    then reconsider i9=-i,j as Element of NAT by INT_1:3;
    p divides -(i*j) by A1,INT_2:10;
    then p divides i9*j;
    then p divides i9 or p divides j by NEWTON:80;
    hence thesis by INT_2:10;
  end;
  suppose
    i<0 & j<0;
    then reconsider i9=-i,j9=-j as Element of NAT by INT_1:3;
    p divides i9*j9 by A1;
    then p divides i9 or p divides j9 by NEWTON:80;
    hence thesis by INT_2:10;
  end;
end;
