reserve

  k,n,m,i,j for Element of NAT,
  K for Field;
reserve L for non empty addLoopStr;
reserve G for non empty multLoopStr;

theorem
  for A being Matrix of n,K st A is invertible holds A@ is
  invertible & (A@)~ =(A~)@
proof
  let A be Matrix of n,K;
  assume A is invertible;
  then consider B being Matrix of n,K such that
A1: B*A=1.(K,n) and
A2: A*B=1.(K,n) by Th19;
  (A*B)@ = (B@)*(A@) by Th30;
  then
A3: (B@)*(A@)=1.(K,n) by A2,MATRIX_6:10;
  (B*A)@ = (A@)*(B@) by Th30;
  then
A4: (A@)*(B@)=1.(K,n) by A1,MATRIX_6:10;
  B=A~ by A1,A2,Th18;
  hence thesis by A3,A4,Th18;
end;
