let z1, z2 be AlgSequence of L; :: thesis: ( ( for i being Nat st i < m holds
z1 . i = eval p,(x |^ i) ) & ( for i being Nat st i >= m holds
z1 . i = 0. L ) & ( for i being Nat st i < m holds
z2 . i = eval p,(x |^ i) ) & ( for i being Nat st i >= m holds
z2 . i = 0. L ) implies z1 = z2 )
assume A12: ( ( for i being Nat st i < m holds
z1 . i = eval p,(x |^ i) ) & ( for i being Nat st i >= m holds
z1 . i = 0. L ) ) ; :: thesis: ( ex i being Nat st
( i < m & not z2 . i = eval p,(x |^ i) ) or ex i being Nat st
( i >= m & not z2 . i = 0. L ) or z1 = z2 )
assume A13: ( ( for i being Nat st i < m holds
z2 . i = eval p,(x |^ i) ) & ( for i being Nat st i >= m holds
z2 . i = 0. L ) ) ; :: thesis: z1 = z2
A14: dom z1 = NAT by FUNCT_2:def 1
.= dom z2 by FUNCT_2:def 1 ;

now

let u be set ; :: thesis: ( u in dom z1 implies z1 . b₁ = z2 . b₁ )
assume u in dom z1 ; :: thesis: z1 . b₁ = z2 . b₁
then reconsider u' = u as Element of NAT by FUNCT_2:def 1;

per cases ( u' < m or m <= u' ) ;

suppose A15: u' < m ; :: thesis: z1 . b₁ = z2 . b₁

hence z1 . u = eval p,(x |^ u') by A12
.= z2 . u by A13, A15 ;
:: thesis: verum

end;

suppose A16: m <= u' ; :: thesis: z1 . b₁ = z2 . b₁

hence z1 . u = 0. L by A12
.= z2 . u by A13, A16 ;
:: thesis: verum

end;

end;

end;

hence z1 = z2 by A14, FUNCT_1:9; :: thesis: verum