reserve x,X for set,
        r,r1,r2,s for Real,
        i,j,k,m,n for Nat;
reserve p,q for Point of TOP-REAL n;

theorem Th12:
  for a,b be Element of F_Real st a = cos r & b = sin r holds
    Det block_diagonal(<*(a,b)][(-b,a),1.(F_Real,n)*>,0.F_Real) = 1.F_Real
  proof
    let a,b be Element of F_Real;
    set A=(a,b)][(-b,a);
    set ONE=1.(F_Real,n);
    set B=block_diagonal(<*A,ONE*>,0.F_Real);
    A1: n=0 or n>=1 by NAT_1:14;
    A2: Det ONE=1_ F_Real or Det ONE=1.F_Real
      by A1,MATRIXR2:41,MATRIX_7:16;
    assume a=cos r & b=sin r;
    then A3: (cos r)*cos r+(sin r)*sin r=a*a-b*(-b)
    .=Det A by MATRIX_9:13;
    A4: cos r=cos.r & sin r=sin.r by SIN_COS:def 17,def 19;
    thus Det B=Det A*Det ONE by MATRIXJ1:52
    .=1.F_Real by A2,A3,A4,SIN_COS:28;
  end;
