set f = proj 1,2;
A1: for x being set st x in REAL holds
ex z being set st
( z in REAL 2 & x = (proj 1,2) . z )

proof

consider y being Element of REAL ;
let x be set ; :: thesis: ( x in REAL implies ex z being set st
( z in REAL 2 & x = (proj 1,2) . z ) )
assume x in REAL ; :: thesis: ex z being set st
( z in REAL 2 & x = (proj 1,2) . z )
then reconsider x1 = x as Element of REAL ;
reconsider z = <*x1,y*> as Element of REAL 2 by FINSEQ_2:121;
(proj 1,2) . z = z . 1 by PDIFF_1:def 1;
then (proj 1,2) . z = x by FINSEQ_1:61;
hence ex z being set st
( z in REAL 2 & x = (proj 1,2) . z ) ; :: thesis: verum

end;

now

let x, y be Element of REAL ; :: thesis: (proj 1,2) . <*x,y*> = x
<*x,y*> is Element of 2 -tuples_on REAL by FINSEQ_2:121;
then (proj 1,2) . <*x,y*> = <*x,y*> . 1 by PDIFF_1:def 1;
hence (proj 1,2) . <*x,y*> = x by FINSEQ_1:61; :: thesis: verum

end;

hence ( dom (proj 1,2) = REAL 2 & rng (proj 1,2) = REAL & ( for x, y being Element of REAL holds (proj 1,2) . <*x,y*> = x ) ) by A1, FUNCT_2:16, FUNCT_2:def 1; :: thesis: verum