reserve i,j,n for Nat,
  K for Field,
  a for Element of K,
  M,M1,M2,M3,M4 for Matrix of n,K;
reserve A for Matrix of K;

theorem Th43:
  for K being Ring
  for M1 being Matrix of n,K
  holds M1 is invertible & (M1@)*M1=1.(K,n) iff M1 is Orthogonal
proof
  let K be Ring;
  let M1 be Matrix of n,K;
A1: width M1=n & len M1=n by MATRIX_0:24;
A2: width (M1@)=n by MATRIX_0:24;
A3: len (M1~)=n by MATRIX_0:24;
A4: width (M1~)=n by MATRIX_0:24;
  thus M1 is invertible & (M1@)*M1=1.(K,n) implies M1 is Orthogonal
  proof
    assume that
A5: M1 is invertible and
A6: (M1@)*M1=1.(K,n);
A7: M1~ is_reverse_of M1 by A5,Def4;
    then (M1~*M1)*M1~=((M1@)*M1)*M1~ by A6;
    then M1~*(M1*M1~)=((M1@)*M1)*M1~ by A1,A4,A3,MATRIX_3:33;
    then M1~*(M1*M1~)=(M1@)*(M1*M1~) by A1,A3,A2,MATRIX_3:33;
    then M1~*(1.(K,n))=(M1@)*(M1*M1~) by A7;
    then M1~*(1.(K,n))=(M1@)*(1.(K,n)) by A7;
    then M1~=(M1@)*(1.(K,n)) by MATRIX_3:19;
    then M1~=M1@ by MATRIX_3:19;
    hence thesis by A5;
  end;
  assume
A8: M1 is Orthogonal;
  then
A9: M1~ is_reverse_of M1 by Def4;
  (M1@)*M1=M1~*M1 by A8;
  hence thesis by A9;
end;
