
theorem Lm12:
for p being Prime
for R being p-characteristic commutative Ring
for a,b being Element of R
for x,y being Element of R|^p st a = x & b = y holds a * b = x * y
proof
let p be Prime;
let F be p-characteristic commutative Ring;
let a,b being Element of F; let x,y be Element of F|^p such that
A1: a = x & b = y;
set M = the set of all a|^p where a is Element of F;
A2: the carrier of F|^p = M by deffp; then
A3: [x,y] in [:M,M:] & [a,b] in [:M,M:] by A1,ZFMISC_1:def 2;
thus x * y = ((the multF of F)||(the carrier of F|^p)).(x,y) by deffp
      .= a * b by A1,A2,A3,FUNCT_1:49;
end;
