reserve X for set,
        n,m,k for Nat,
        K for Field,
        f for n-element real-valued FinSequence,
        M for Matrix of n,m,F_Real;

theorem Th52:
  for n be non empty Nat holds
    RealFuncExtMult(Seg n) = n -Mult_over F_Real
  proof
    let n be non empty Nat;

    set OP1 = RealFuncExtMult Seg n;
    set OP2 = n -Mult_over F_Real;

    A1: Funcs(Seg n,REAL) = REAL n by FINSEQ_2:93;
    for x be Element of REAL,
        y be Element of REAL n
    holds OP1.(x,y) = OP2.(x,y)
    proof
      let x be Element of REAL,
          y be Element of REAL n;
      reconsider y0 = y as Element of Funcs(Seg n,REAL) by FINSEQ_2:93;
      [x,y] in [:REAL,REAL n:] by ZFMISC_1:87; then
      OP2.(x,y) in REAL n by FUNCT_2:5; then
      reconsider h = OP2.(x,y0) as Element of Funcs(Seg n,REAL)
        by FINSEQ_2:93;

      for i be Element of Seg n
      holds h.i = x * (y0.i)
      proof
        let i be Element of Seg n;
        A2: OP2.[x,y0]
          = OP2.(x,y0)
         .= (multreal) [;] (x,y0) by PRVECT_1:def 4;
        dom h = Seg n by FUNCT_2:def 1;

        hence h.i = (multreal).(x,y0.i) by FUNCOP_1:32,A2
           .= x*(y0.i) by BINOP_2:def 11;
      end;
      hence thesis by FUNCSDOM:4;
    end;
    hence thesis by BINOP_1:2,A1;
  end;
