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 Th14:
  1 <= i & i < j & j <= n & k in Seg n & k <> i & k <> j implies
    @p"*"Col(Rotation(i,j,n,r),k) = p.k
proof
  set S=Seg n;
  assume that
  A1: 1<=i & i<j & j<=n and
  A2: k in Seg n and
  A3: k<>i & k<>j;
  set O=Rotation(i,j,n,r),C=Col(O,k);
  A4: Indices O=[:S,S:] by MATRIX_0:24;
  then A5: [k,k] in Indices O by A2,ZFMISC_1:87;
  A6: len O=n by MATRIX_0:25;
  then A7: dom O=S by FINSEQ_1:def 3;
  len C=n by A6,MATRIX_0:def 8;
  then A8: dom C=S by FINSEQ_1:def 3;
  A9: now let m such that
       A10: m in dom C and
       A11: m<>k;
       A12: [m,k] in Indices O by A2,A4,A8,A10,ZFMISC_1:87;
       not k in {i,j} by A3,TARSKI:def 2;
       then A13: {m,k}<>{i,j} by TARSKI:def 2;
       thus C.m=O*(m,k) by A7,A8,A10,MATRIX_0:def 8
       .=0.F_Real by A1,A11,A12,A13,Def3;
     end;
  len p=n by CARD_1:def 7;
  then A14: dom p=S by FINSEQ_1:def 3;
  C.k=O*(k,k) by A2,A7,MATRIX_0:def 8
   .=1.F_Real by A1,A3,A5,Def3;
  hence p.k=Sum(mlt(C,@p)) by A2,A8,A9,A14,MATRIX_3:17
   .=@p"*"C by FVSUM_1:64;
end;
