let f1, f2 be Function of [: the carrier of (), the carrier of ():], the carrier of F_Real; :: thesis: ( ( for v, u being Vector of L
for vv, uu being Vector of () st vv = () . v & uu = () . u holds
f1 . (vv,uu) = <;v,u;> ) & ( for v, u being Vector of L
for vv, uu being Vector of () st vv = () . v & uu = () . u holds
f2 . (vv,uu) = <;v,u;> ) implies f1 = f2 )

assume AS1: for v, u being Vector of L
for vv, uu being Vector of () st vv = () . v & uu = () . u holds
f1 . (vv,uu) = <;v,u;> ; :: thesis: ( ex v, u being Vector of L ex vv, uu being Vector of () st
( vv = () . v & uu = () . u & not f2 . (vv,uu) = <;v,u;> ) or f1 = f2 )

assume AS2: for v, u being Vector of L
for vv, uu being Vector of () st vv = () . v & uu = () . u holds
f2 . (vv,uu) = <;v,u;> ; :: thesis: f1 = f2
for x being Element of [: the carrier of (), the carrier of ():] holds f1 . x = f2 . x
proof
let x be Element of [: the carrier of (), the carrier of ():]; :: thesis: f1 . x = f2 . x
consider vv, uu being object such that
B0: ( vv in the carrier of () & uu in the carrier of () & x = [vv,uu] ) by ZFMISC_1:def 2;
reconsider vv = vv, uu = uu as Vector of () by B0;
consider v being Vector of L such that
B2: (MorphsZQ L) . v = vv by ZMODUL08:22;
consider u being Vector of L such that
B3: (MorphsZQ L) . u = uu by ZMODUL08:22;
thus f1 . x = f1 . (vv,uu) by B0
.= <;v,u;> by AS1, B2, B3
.= f2 . (vv,uu) by AS2, B2, B3
.= f2 . x by B0 ; :: thesis: verum
end;
hence f1 = f2 ; :: thesis: verum