defpred S₁[ object , object , object ] means ex r being Element of REAL ex g being Point of G ex f being Point of F st
( r = $1 & $2 = [g,f] & $3 = [(r * g),(r * f)] );
set CarrG = the carrier of G;
set CarrF = the carrier of F;
A1: for x, y being object st x in REAL & y in [: the carrier of G, the carrier of F:] holds
ex z being object st
( z in [: the carrier of G, the carrier of F:] & S₁[x,y,z] )

proof

let x, y be object ; :: thesis: ( x in REAL & y in [: the carrier of G, the carrier of F:] implies ex z being object st
( z in [: the carrier of G, the carrier of F:] & S₁[x,y,z] ) )
assume A2: ( x in REAL & y in [: the carrier of G, the carrier of F:] ) ; :: thesis: ex z being object st
( z in [: the carrier of G, the carrier of F:] & S₁[x,y,z] )
then reconsider r = x as Element of REAL ;
consider y1 being Point of G, y2 being Point of F such that
A3: y = [y1,y2] by A2, Lm1;
set z = [(r * y1),(r * y2)];
( [(r * y1),(r * y2)] in [: the carrier of G, the carrier of F:] & S₁[x,y,[(r * y1),(r * y2)]] ) by A3;
hence ex z being object st
( z in [: the carrier of G, the carrier of F:] & S₁[x,y,z] ) ; :: thesis: verum

end;

consider MLTGF being Function of [:REAL,[: the carrier of G, the carrier of F:]:],[: the carrier of G, the carrier of F:] such that
A4: for x, y being object st x in REAL & y in [: the carrier of G, the carrier of F:] holds
S₁[x,y,MLTGF . (x,y)] from BINOP_1:sch 1(A1);

now :: thesis: for r being Real
for g being Point of G
for f being Point of F holds MLTGF . (r,[g,f]) = [(r * g),(r * f)]

let r be Real; :: thesis: for g being Point of G
for f being Point of F holds MLTGF . (r,[g,f]) = [(r * g),(r * f)]
let g be Point of G; :: thesis: for f being Point of F holds MLTGF . (r,[g,f]) = [(r * g),(r * f)]
let f be Point of F; :: thesis: MLTGF . (r,[g,f]) = [(r * g),(r * f)]
reconsider rr = r as Element of REAL by XREAL_0:def 1;
S₁[rr,[g,f],MLTGF . (rr,[g,f])] by A4;
then consider rr being Element of REAL , gg being Point of G, ff being Point of F such that
A5: ( rr = r & [g,f] = [gg,ff] & MLTGF . (r,[g,f]) = [(rr * gg),(r * ff)] ) ;
( g = gg & f = ff ) by A5, XTUPLE_0:1;
hence MLTGF . (r,[g,f]) = [(r * g),(r * f)] by A5; :: thesis: verum

end;

hence ex b₁ being Function of [:REAL,[: the carrier of G, the carrier of F:]:],[: the carrier of G, the carrier of F:] st
for r being Real
for g being Point of G
for f being Point of F holds b₁ . (r,[g,f]) = [(r * g),(r * f)] ; :: thesis: verum