set f = proj 1,3;
A1: for x being set st x in REAL holds
ex u being set st
( u in REAL 3 & x = (proj 1,3) . u )

proof

let x be set ; :: thesis: ( x in REAL implies ex u being set st
( u in REAL 3 & x = (proj 1,3) . u ) )
assume x in REAL ; :: thesis: ex u being set st
( u in REAL 3 & x = (proj 1,3) . u )
then reconsider x1 = x as Element of REAL ;
consider y, z being Element of REAL ;
reconsider u = <*x1,y,z*> as Element of REAL 3 by FINSEQ_2:124;
(proj 1,3) . u = u . 1 by PDIFF_1:def 1;
then (proj 1,3) . u = x by FINSEQ_1:62;
hence ex u being set st
( u in REAL 3 & x = (proj 1,3) . u ) ; :: thesis: verum

end;

now

let x, y, z be Element of REAL ; :: thesis: (proj 1,3) . <*x,y,z*> = x
<*x,y,z*> is Element of 3 -tuples_on REAL by FINSEQ_2:124;
then (proj 1,3) . <*x,y,z*> = <*x,y,z*> . 1 by PDIFF_1:def 1;
hence (proj 1,3) . <*x,y,z*> = x by FINSEQ_1:62; :: thesis: verum

end;

hence ( dom (proj 1,3) = REAL 3 & rng (proj 1,3) = REAL & ( for x, y, z being Element of REAL holds (proj 1,3) . <*x,y,z*> = x ) ) by A1, FUNCT_2:16, FUNCT_2:def 1; :: thesis: verum