let R be non empty right_add-cancelable left_zeroed distributive add-associative right_zeroed doubleLoopStr ; :: thesis: for a, b being Element of R
for n being Element of NAT holds b * (n * a) = (b * a) * n
let a, b be Element of R; :: thesis: for n being Element of NAT holds b * (n * a) = (b * a) * n
let n be Element of NAT ; :: thesis: b * (n * a) = (b * a) * n
defpred S₁[ Element of NAT ] means b * ($1 * a) = (b * a) * $1;

A1: now

let k be Element of NAT ; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A2: S₁[k] ; :: thesis: S₁[k + 1]
b * ((k + 1) * a) = b * (a + (k * a)) by Def6
.= (b * a) + ((b * a) * k) by A2, VECTSP_1:def 11
.= ((b * a) * 1) + ((b * a) * k) by Th15
.= (b * a) * (k + 1) by Th17 ;
hence S₁[k + 1] ; :: thesis: verum

end;

b * (0 * a) = b * (0. R) by Def6
.= 0. R by Th2
.= (b * a) * 0 by Def7 ;
then A3: S₁[ 0 ] ;
for n being Element of NAT holds S₁[n] from NAT_1:sch 1(A3, A1);
hence b * (n * a) = (b * a) * n ; :: thesis: verum