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 R being commutative Ring, M1,M2 being Matrix of n,R
  holds n > 0 & M1 is Orthogonal & M2 is Orthogonal implies 
    M1~*M2 is Orthogonal
proof
  let R be commutative Ring;
  let M1,M2 be Matrix of n,R;
  assume that
A1: n > 0 and
A2: M1 is Orthogonal and
A3: M2 is Orthogonal;
A4: M1 is invertible by A2; then
A5: M1~ is invertible;
A6: M2 is invertible by A3;
  then
A7: M1~*M2 is invertible by A5,Th37;
  (M1~*M2)~=M2~*(M1~)~ by A6,A5,Th37;
  then
A8: (M1~*M2)~=M2~*M1 by A4,Th17;
A9: len M2=n by MATRIX_0:24;
A10: width (M1~)=n & width M2=n by MATRIX_0:24;
A11: width M1=n & len M1=n by MATRIX_0:24;
  M1@=M1~ & M2@=M2~ by A2,A3;
  then (M1~*M2)@=(M2~)*((M1@)@) by A1,A10,A9,MATRIX_3:22
    .=(M2~)*M1 by A1,A11,MATRIX_0:57;
  hence thesis by A7,A8;
end;
