let D be non empty set ; :: thesis: for b being BinOp of D
for d, d1, d2 being Element of D holds b "**" <%d,d1,d2%> = b . ((b . (d,d1)),d2)
let b be BinOp of D; :: thesis: for d, d1, d2 being Element of D holds b "**" <%d,d1,d2%> = b . ((b . (d,d1)),d2)
let d, d1, d2 be Element of D; :: thesis: b "**" <%d,d1,d2%> = b . ((b . (d,d1)),d2)
set F = <%d,d1,d2%>;
len <%d,d1,d2%> = 3 by AFINSQ_1:39;
then consider f being sequence of D such that
A1: f . 0 = <%d,d1,d2%> . 0 and
A2: for n being Nat st n + 1 < 3 holds
f . (n + 1) = b . ((f . n),(<%d,d1,d2%> . (n + 1))) and
A3: b "**" <%d,d1,d2%> = f . (3 - 1) by Def8;
A4: f . (1 + 1) = b . ((f . 1),(<%d,d1,d2%> . (1 + 1))) by A2;
f . (zz + 1) = b . ((f . zz),(<%d,d1,d2%> . (zz + 1))) by A2;
hence b "**" <%d,d1,d2%> = b . ((b . (d,d1)),d2) by A1, A3, A4; :: thesis: verum