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,M3 being Matrix of n,R
  holds n > 0 & M1 is Orthogonal & M2 is Orthogonal & M3 is Orthogonal implies
  M1*M2*M3 is Orthogonal
proof
  let R be commutative Ring;
  let M1,M2,M3 be Matrix of n,R;
  assume that
A1: n > 0 and
A2: M1 is Orthogonal & M2 is Orthogonal and
A3: M3 is Orthogonal;
A4: M3 is invertible by A3;
  set M5=M3~*M2~*M1~;
  set M4=M1*M2*M3;
A5: width M1=n & len M2=n by MATRIX_0:24;
  M1 is invertible & M2 is invertible by A2;
  then
A6: M4~=M5 & M4 is invertible by A4,Th57;
A7: width M2=n & M3@=M3~ by A3,MATRIX_0:24;
A8: width (M2~)=n & width (M3~)=n by MATRIX_0:24;
A9: M1@=M1~ & M2@=M2~ by A2;
A10: width (M1*M2)=n by MATRIX_0:24;
A11: len (M1~)=n & len (M2~)=n by MATRIX_0:24;
  len M3=n & width M3=n by MATRIX_0:24;
  then (M1*M2*M3)@ = M3@*((M1*M2)@) by A1,A10,MATRIX_3:22
    .=M3~*(M2~*M1~) by A1,A5,A9,A7,MATRIX_3:22
    .=M5 by A8,A11,MATRIX_3:33;
  hence thesis by A6;
end;
