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 Th18:
 1 <= i & i < j & j <= n implies Rotation(i,j,n,0) = 1.(F_Real,n)
proof
  set O=Rotation(i,j,n,0);
  assume A1: 1<=i & i<j & j<=n;
  A2: for k,m st[k,m] in Indices O & k<>m holds O*(k,m)=0.F_Real
  proof
    let k,m;
    assume that
    A3: [k,m] in Indices O and
    A4: k<>m;
    per cases;
    suppose k=i & m=j or k=j & m=i;
      then O*(k,m)=sin 0 or O*(k,m)=-sin 0 by A1,Def3;
      hence thesis by SIN_COS:31;
    end;
    suppose k=i & m<>j;
      then not m in {i,j} by A4,TARSKI:def 2;
      then {k,m}<>{i,j} by TARSKI:def 2;
      hence thesis by A1,A3,A4,Def3;
    end;
    suppose k=j & m<>i;
      then not m in {i,j} by A4,TARSKI:def 2;
      then {k,m}<>{i,j} by TARSKI:def 2;
      hence thesis by A1,A3,A4,Def3;
    end;
    suppose m=i & k<>j;
      then not k in {i,j} by A4,TARSKI:def 2;
      then {k,m}<>{i,j} by TARSKI:def 2;
      hence thesis by A1,A3,A4,Def3;
    end;
    suppose m=j & k<>i;
      then not k in {i,j} by A4,TARSKI:def 2;
      then {k,m}<>{i,j} by TARSKI:def 2;
      hence thesis by A1,A3,A4,Def3;
    end;
    suppose k<>i & k<>j & m<>i & m<>j;
      then not m in {i,j} by TARSKI:def 2;
      then {k,m}<>{i,j} by TARSKI:def 2;
      hence thesis by A1,A3,A4,Def3;
    end;
  end;
  for k st[k,k] in Indices O holds O*(k,k)=1.F_Real
  proof
    let k;
    assume A5: [k,k] in Indices O;
    k=i or k=j or k<>i & k<>j;
    hence thesis by A1,A5,Def3,SIN_COS:31;
  end;
  hence thesis by A2,MATRIX_1:def 3;
end;
