defpred S₁[ Real, Real, set ] means ex c being Element of REAL 2 st
( c = $3 & $3 = <*$1,$2*> );
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 S₁[x,y,u]

proof

let x, y be Element of REAL ; :: thesis: ex u being Element of REAL 2 st S₁[x,y,u]
take <*x,y*> ; :: thesis: ( <*x,y*> is Element of REAL 2 & S₁[x,y,<*x,y*>] )
thus <*x,y*> is Element of REAL 2 by FINSEQ_2:137; :: thesis: S₁[x,y,<*x,y*>]
<*x,y*> in 2 -tuples_on REAL by FINSEQ_2:137;
hence S₁[x,y,<*x,y*>] ; :: thesis: verum

end;

consider f being Function of [:REAL,REAL:],(REAL 2) such that
A3: for x, y being Element of REAL holds S₁[x,y,f . (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 f = f as Function of [:R^1,R^1:],(TOP-REAL 2) by A1, TOPMETR:17;
take f ; :: according to T_0TOPSP:def 1 :: thesis: f is being_homeomorphism
for x, y being Real holds f . [x,y] = <*x,y*>

proof

let x, y be Real; :: thesis: f . [x,y] = <*x,y*>
( x in REAL & y in REAL ) by XREAL_0:def 1;
then S₁[x,y,f . (x,y)] by A3;
hence f . [x,y] = <*x,y*> ; :: thesis: verum

end;

hence f is being_homeomorphism by Th69; :: thesis: verum