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 Th20:
  1 <= i & i < j & j <= n & k <> i & k <> j implies
   (Mx2Tran Rotation(i,j,n,r)).p.k=p.k
proof
  set O=Rotation(i,j,n,r),M=Mx2Tran O,Mp=M.p,S=Seg n;
  assume A1: 1<=i & i<j & j<=n & k<>i & k<>j;
  len Mp=n by CARD_1:def 7;
  then A2: dom Mp=S by FINSEQ_1:def 3;
  len p=n by CARD_1:def 7;
  then A3: dom p=S by FINSEQ_1:def 3;
  per cases;
    suppose A4: k in S;
      then 1<=k & k<=n by FINSEQ_1:1;
      hence Mp.k=@p"*"Col(O,k) by MATRTOP1:18
      .=p.k by A1,A4,Th14;
    end;
    suppose A5: not k in S;
      then Mp.k={} by A2,FUNCT_1:def 2;
      hence thesis by A3,A5,FUNCT_1:def 2;
    end;
end;
