set Z = EMbedding L;
set D = DivisibleMod L;
let f1, f2 be Function of [: the carrier of (), the carrier of ():],F_Real; :: thesis: ( ( for vv, uu being Vector of ()
for v, u being Vector of ()
for a, b being Element of INT.Ring
for ai, bi being Element of F_Real st a = ai & b = bi & ai <> 0 & bi <> 0 & v = a * vv & u = b * uu holds
f1 . (vv,uu) = ((ai ") * (bi ")) * (() . (v,u)) ) & ( for vv, uu being Vector of ()
for v, u being Vector of ()
for a, b being Element of INT.Ring
for ai, bi being Element of F_Real st a = ai & b = bi & ai <> 0 & bi <> 0 & v = a * vv & u = b * uu holds
f2 . (vv,uu) = ((ai ") * (bi ")) * (() . (v,u)) ) implies f1 = f2 )

assume AS1: for vv, uu being Vector of ()
for v, u being Vector of ()
for a, b being Element of INT.Ring
for ai, bi being Element of F_Real st a = ai & b = bi & ai <> 0 & bi <> 0 & v = a * vv & u = b * uu holds
f1 . (vv,uu) = ((ai ") * (bi ")) * (() . (v,u)) ; :: thesis: ( ex vv, uu being Vector of () ex v, u being Vector of () ex a, b being Element of INT.Ring ex ai, bi being Element of F_Real st
( a = ai & b = bi & ai <> 0 & bi <> 0 & v = a * vv & u = b * uu & not f2 . (vv,uu) = ((ai ") * (bi ")) * (() . (v,u)) ) or f1 = f2 )

assume AS2: for vv, uu being Vector of ()
for v, u being Vector of ()
for a, b being Element of INT.Ring
for ai, bi being Element of F_Real st a = ai & b = bi & ai <> 0 & bi <> 0 & v = a * vv & u = b * uu holds
f2 . (vv,uu) = ((ai ") * (bi ")) * (() . (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 a being Element of INT.Ring such that
B1: ( a <> 0 & a * vv in EMbedding L ) by ZMODUL08:29;
consider b being Element of INT.Ring such that
B2: ( b <> 0 & b * uu in EMbedding L ) by ZMODUL08:29;
reconsider v = a * vv as Vector of () by B1;
reconsider u = b * uu as Vector of () by B2;
INT c= the carrier of F_Real by NUMBERS:5;
then reconsider ai = a, bi = b as Element of F_Real ;
thus f1 . x = f1 . (vv,uu) by B0
.= ((ai ") * (bi ")) * (() . (v,u)) by AS1, B1, B2
.= f2 . (vv,uu) by AS2, B1, B2
.= f2 . x by B0 ; :: thesis: verum
end;
hence f1 = f2 ; :: thesis: verum