
theorem Th20:
  ex f being Function of [:R^1,R^1:], TOP-REAL 2 st for x, y being
  Real holds f. [x,y] = <*x,y*>
proof
  defpred P[Element of REAL,Element of REAL,set] means ex c being Element of
  REAL 2 st c = $3 & $3 = <*$1,$2*>;
A1: 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*>;
    <*x,y*> is Element of REAL 2 by FINSEQ_2:137;
    hence thesis;
  end;
  consider f being Function of [:REAL,REAL:],REAL 2 such that
A2: for x, y being Element of REAL holds P[x,y,f.(x,y)] from BINOP_1:sch
  3(A1);
  the carrier of [:R^1,R^1:] = [:the carrier of R^1,the carrier of R^1:]
  by BORSUK_1:def 2;
  then reconsider f as Function of [:R^1,R^1:], TOP-REAL 2 by EUCLID:22
,TOPMETR:17;
  take f;
  for x, y being Real holds f. [x,y] = <*x,y*>
  proof
    let x, y be Real;
     reconsider x,y as Element of REAL by XREAL_0:def 1;
    P[x,y,f.(x,y)] by A2;
    hence thesis;
  end;
  hence thesis;
end;
