reserve a, b for Real,
  r for Real,
  rr for Real,
  i, j, n for Nat,
  M for non empty MetrSpace,
  p, q, s for Point of TOP-REAL 2,
  e for Point of Euclid 2,
  w for Point of Euclid n,
  z for Point of M,
  A, B for Subset of TOP-REAL n,
  P for Subset of TOP-REAL 2,
  D for non empty Subset of TOP-REAL 2;
reserve a, b for Real;
reserve a, b for Real;
reserve r for Real;

theorem Th71:
  for A, B being compact Subset of REAL holds product ((1,2) --> (
  A,B)) is compact Subset of TOP-REAL 2
proof
  defpred P[Real,Real,set] means ex c being Element of
  REAL 2 st c = $3 & $3 = <*$1,$2*>;
  let A, B be compact Subset of REAL;
  reconsider X = product ((1,2) --> (A,B)) as Subset of TOP-REAL 2 by Th20;
  reconsider A1 = A, B1 = B as Subset of R^1 by TOPMETR:17;
A1: the carrier of TOP-REAL 2 = REAL 2 by EUCLID:22;
A2: for x, y being Element of REAL ex u being Element of REAL 2 st P[x,y,u]
  proof
    let x, y be Element of REAL;
    take <*x,y*>;
    thus <*x,y*> is Element of REAL 2 by FINSEQ_2:137;
    <*x,y*> in 2-tuples_on REAL by FINSEQ_2:137;
    hence thesis;
  end;
  consider h being Function of [:REAL,REAL:],REAL 2 such that
A3: for x, y being Element of REAL holds P[x,y,h.(x,y)] from BINOP_1:sch
  3(A2);
  the carrier of [:R^1,R^1:] = [:the carrier of R^1,the carrier of R^1 :]
  by BORSUK_1:def 2;
  then reconsider h as Function of [:R^1,R^1:], TOP-REAL 2 by A1,TOPMETR:17;
A4: for x, y being Real holds h. [x,y] = <*x,y*>
  proof
    let x, y be Real;
    x in REAL & y in REAL by XREAL_0:def 1;
    then P[x,y,h.(x,y)] by A3;
    hence thesis;
  end;
  then
A5: h is being_homeomorphism by Th69;
A6: B1 is compact by JORDAN5A:25;
  A1 is compact by JORDAN5A:25;
  then
A7: [:A1,B1:] is compact by A6,BORSUK_3:23;
  h.:[:A1,B1:] = X by A4,Th68;
  hence thesis by A7,A5,WEIERSTR:8;
end;
