A2:
( [:REAL ,V1:] c= [:REAL ,the carrier of V:] & dom the Mult of V = [:REAL ,the carrier of V:] )
by FUNCT_2:def 1, ZFMISC_1:118;
A3:
for z being set st z in [:REAL ,V1:] holds
(the Mult of V | [:REAL ,V1:]) . z in V1
proof
let z be
set ;
( z in [:REAL ,V1:] implies (the Mult of V | [:REAL ,V1:]) . z in V1 )
assume A4:
z in [:REAL ,V1:]
;
(the Mult of V | [:REAL ,V1:]) . z in V1
consider r,
x being
set such that A5:
r in REAL
and A6:
x in V1
and A7:
z = [r,x]
by A4, ZFMISC_1:def 2;
reconsider r =
r as
Real by A5;
reconsider y =
x as
VECTOR of
V by A6;
[r,x] in dom (the Mult of V | [:REAL ,V1:])
by A2, A4, A7, RELAT_1:91;
then
(the Mult of V | [:REAL ,V1:]) . z = r * y
by A7, FUNCT_1:70;
hence
(the Mult of V | [:REAL ,V1:]) . z in V1
by A1, A6, Def10;
verum
end;
dom (the Mult of V | [:REAL ,V1:]) = [:REAL ,V1:]
by A2, RELAT_1:91;
hence
the Mult of V | [:REAL ,V1:] is Function of [:REAL ,V1:],V1
by A3, FUNCT_2:5; verum