
theorem ch0a:
for F being Field,
    a being Element of F, b being non zero Element of F
for i being Integer
st i '*' a <> 0.F & i '*' b <> 0.F holds (i '*' a) * ((i '*' b)") = a * b"
proof
let F be Field, a be Element of F, b be non zero Element of F;
let i be Integer;
assume AS: i '*' a <> 0.F & i '*' b <> 0.F;
H: b <> 0.F;
((a * b") * ((i '*' a)")) * (i '*' b)
     = (((i '*' a)") * (a * b")) * (i '*' b) by GROUP_1:def 12
    .= ((i '*' a)") * ((a * b") * (i '*' b)) by GROUP_1:def 3
    .= ((i '*' a)") * ((i '*' b) * (a * b")) by GROUP_1:def 12
    .= ((i '*' a)") * (i '*' (b * (a * b"))) by REALALG2:5
    .= ((i '*' a)") * (i '*' (b * (b" * a))) by GROUP_1:def 12
    .= ((i '*' a)") * (i '*' ((b * b") * a)) by GROUP_1:def 3
    .= ((i '*' a)") * (i '*' ((b" * b) * a)) by GROUP_1:def 12
    .= ((i '*' a)") * (i '*' (1.F * a)) by H,VECTSP_1:def 10
    .= 1.F by AS,VECTSP_1:def 10;
then (a * b") * ((i '*' a)") = (i '*' b)" by AS,VECTSP_1:def 10;
hence (i '*' a) * ((i '*' b)")
      = (i '*' a) * ((i '*' a)" * (a * b")) by GROUP_1:def 12
     .= ((i '*' a) * (i '*' a)") * (a * b") by GROUP_1:def 3
     .= ((i '*' a)" * (i '*' a)) * (a * b") by GROUP_1:def 12
     .= 1.F * (a * b") by AS,VECTSP_1:def 10
     .= a * b";
end;
