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
  for K being commutative Ring
  for M1,M2 being Matrix of n,K holds
  n > 0 & M1 is Orthogonal & M2 is Orthogonal implies M1*M2 is Orthogonal
proof
  let K be commutative Ring;
  let M1,M2 be Matrix of n,K;
  assume that
A1: n > 0 and
A2: M1 is Orthogonal & M2 is Orthogonal;
  M1 is invertible & M2 is invertible by A2;
  then
A3: M1*M2 is invertible & (M1*M2)~=M2~*M1~ by Th46;
A4: width M2=n by MATRIX_0:24;
A5: width M1=n & len M2=n by MATRIX_0:24;
  M1@=M1~ & M2@=M2~ by A2;
  then (M1*M2)@ = M2~*M1~ by A1,A5,A4,MATRIX_3:22;
  hence thesis by A3;
end;
