theorem :: PRVECT_4:8

for E, F, G being RealLinearSpace holds
( ( for x being set holds
( x is Point of (product <*E,F,G*>) iff ex x1 being Point of E ex x2 being Point of F ex x3 being Point of G st x = <*x1,x2,x3*> ) ) & ( for x1, y1 being Point of E
for x2, y2 being Point of F
for x3, y3 being Point of G holds <*x1,x2,x3*> + <*y1,y2,y3*> = <*(x1 + y1),(x2 + y2),(x3 + y3)*> ) & 0. (product <*E,F,G*>) = <*(0. E),(0. F),(0. G)*> & ( for x1 being Point of E
for x2 being Point of F
for x3 being Point of G holds - <*x1,x2,x3*> = <*(- x1),(- x2),(- x3)*> ) & ( for x1 being Point of E
for x2 being Point of F
for x3 being Point of G
for a being Real holds a * <*x1,x2,x3*> = <*(a * x1),(a * x2),(a * x3)*> ) )