let L be Field; :: thesis: for x being Element of L st x <> 0. L holds
for i being Integer holds pow ((x "),i) = (pow (x,i)) "
let x be Element of L; :: thesis: ( x <> 0. L implies for i being Integer holds pow ((x "),i) = (pow (x,i)) " )
assume A1: x <> 0. L ; :: thesis: for i being Integer holds pow ((x "),i) = (pow (x,i)) "
let i be Integer; :: thesis: pow ((x "),i) = (pow (x,i)) "

per cases ( i >= 0 or i < 0 ) ;

suppose i >= 0 ; :: thesis: pow ((x "),i) = (pow (x,i)) "

then reconsider n = i as Element of NAT by INT_1:3;
thus pow ((x "),i) = (pow (x,n)) " by A1, Lm9
.= (pow (x,i)) " ; :: thesis: verum

end;

suppose A2: i < 0 ; :: thesis: pow ((x "),i) = (pow (x,i)) "

A3: pow (x,|.i.|) = x |^ |.i.| by Def2;
thus pow ((x "),i) = (pow ((x "),|.i.|)) " by A2, Lm3
.= pow (((x ") "),|.i.|) by A1, Lm9, VECTSP_1:25
.= pow (x,|.i.|) by A1, VECTSP_1:24
.= ((pow (x,|.i.|)) ") " by A1, A3, Th1, VECTSP_1:24
.= (pow (x,i)) " by A2, Lm3 ; :: thesis: verum

end;

end;