reserve X for set;
reserve a,b,c,k,m,n for Nat;
reserve i,j for Integer;
reserve r,s for Real;
reserve p,p1,p2,p3 for Prime;

theorem
  { [x,y] where x,y is Integer: 2*x|^3 + x*y - 7 = 0 } =
  { [1,5], [7,-97], [-1,-9], [-7,-99] }
  proof
    set A = { [x,y] where x,y is Integer: 2*x|^3 + x*y - 7 = 0 };
    set B = { [1,5], [7,-97], [-1,-9], [-7,-99] };
    thus A c= B
    proof
      let a be object;
      assume a in A;
      then consider x,y being Integer such that
A1:   a = [x,y] and
A2:   2*x|^3 + x*y - 7 = 0;
A3:   x|^3 = x*x*x by POLYEQ_5:2;
      then x*(2*x*x+y) = 7 by A2;
      then x divides 7;
      then x = 1 or x = 7 or x = -1 or x = -7 by Th94,XPRIMES1:7;
      hence thesis by A1,A2,A3,ENUMSET1:def 2;
    end;
    let a be object;
    assume a in B;
    then per cases by ENUMSET1:def 2;
    suppose
A4:   a = [1,5];
      2*1|^3 + 1*5 - 7 = 0;
      hence thesis by A4;
    end;
    suppose
A5:   a = [7,-97];
      7|^3 = 7*7*7 by POLYEQ_5:2;
      then 2*7|^3 + 7*(-97) - 7 = 0;
      hence thesis by A5;
    end;
    suppose
A6:   a = [-1,-9];
      (-1)|^3 = (-1)*(-1)*(-1) by POLYEQ_5:2;
      then 2*(-1)|^3 + (-1)*(-9) - 7 = 0;
      hence thesis by A6;
    end;
    suppose
A7:   a = [-7,-99];
      (-7)|^3 = (-7)*(-7)*(-7) by POLYEQ_5:2;
      then 2*(-7)|^3 + (-7)*(-99) - 7 = 0;
      hence thesis by A7;
    end;
  end;
