consider M being Matrix of n,F_Real such that
A2: Det M = - (1. F_Real) and
A3: ( M * (i,i) = - (1. F_Real) & ( for k, m being Nat st [k,m] in Indices M holds
( ( k = m & k <> i implies M * (k,k) = 1. F_Real ) & ( k <> m implies M * (k,m) = 0. F_Real ) ) ) ) by A1, Lm1;
Det M <> 0. F_Real by A2;
then M is invertible by LAPLACE:34;
hence ex b₁ being invertible Matrix of n,F_Real st
( b₁ * (i,i) = - (1. F_Real) & ( for k, m being Nat st [k,m] in Indices b₁ holds
( ( k = m & k <> i implies b₁ * (k,k) = 1. F_Real ) & ( k <> m implies b₁ * (k,m) = 0. F_Real ) ) ) ) by A3; :: thesis: verum