let x be Real; :: thesis: for p being Point of (TOP-REAL 3) holds
( (x * p) `1 = x * (p `1) & (x * p) `2 = x * (p `2) & (x * p) `3 = x * (p `3) )
let p be Point of (TOP-REAL 3); :: thesis: ( (x * p) `1 = x * (p `1) & (x * p) `2 = x * (p `2) & (x * p) `3 = x * (p `3) )
A1: (x * p) `3 = |[(x * (p `1)),(x * (p `2)),(x * (p `3))]| `3 by Th7;
( (x * p) `1 = |[(x * (p `1)),(x * (p `2)),(x * (p `3))]| `1 & (x * p) `2 = |[(x * (p `1)),(x * (p `2)),(x * (p `3))]| `2 ) by Th7;
hence ( (x * p) `1 = x * (p `1) & (x * p) `2 = x * (p `2) & (x * p) `3 = x * (p `3) ) by A1; :: thesis: verum