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 Th13:
  1 <= i & i < j & j <= n implies Det Rotation(i,j,n,r)=1.F_Real
proof
  assume A1: 1<=i & i<j & j<=n;
  then consider A be Matrix of n,F_Real such that
  A2: Det A=1.F_Real and
  A3: A*(i,i)=cos r & A*(j,j)=cos r & A*(i,j)=sin r & A*(j,i)=-sin r &
  for k,m st[k,m] in Indices A holds
     (k=m & k<>i & k<>j implies A*(k,k)=1.F_Real) &
     (k<>m & {k,m}<>{i,j} implies A*(k,m)=0.F_Real) by Lm3;
  Det A<>0.F_Real by A2;
  then A is invertible by LAPLACE:34;
  hence thesis by A1,A2,A3,Def3;
end;
