reserve n   for Nat,
        r,s for Real,
        x,y for Element of REAL n,
        p,q for Point of TOP-REAL n,
        e   for Point of Euclid n;

theorem Th13:
  for f being n-element FinSequence of [:REAL,REAL:] holds
  ex x being Element of [:REAL n,REAL n:] st for i being Nat st
  i in Seg n holds (x`1).i = (f/.i)`1 & (x`2).i = (f/.i)`2
  proof
    let f be n-element FinSequence of [:REAL,REAL:];
      defpred P[Nat,set] means $2 = (f/.$1)`1;
A2:   for i be Nat st i in Seg n ex d be Element of REAL st P[i,d];
      ex x1 being FinSequence of REAL st len x1 = n & for i be Nat st
        i in Seg n
      holds P[i,x1/.i] from FINSEQ_4:sch 1(A2);
      then consider x1 being FinSequence of REAL such that
A3:   len x1 = n and
A4:   for i be Nat st i in Seg n holds x1/.i = (f/.i)`1;
      dom x1 = Seg n by A3,FINSEQ_1:def 3;
      then reconsider x1 as Element of REAL n by REAL_NS1:6;
      defpred Q[Nat,set] means $2 = (f/.$1)`2;
A5:   for i be Nat st i in Seg n ex d be Element of REAL st Q[i,d];
      ex x2 being FinSequence of REAL st len x2 = n & for i be Nat st
        i in Seg n
      holds Q[i,x2/.i] from FINSEQ_4:sch 1(A5);
      then consider x2 being FinSequence of REAL such that
A6:   len x2 = n and
A7:   for i be Nat st i in Seg n holds x2/.i = (f/.i)`2;
      dom x2 = Seg n by A6,FINSEQ_1:def 3;
      then reconsider x2 as Element of REAL n by REAL_NS1:6;
      reconsider x = [x1,x2] as Element of [:REAL n,REAL n:] by ZFMISC_1:def 2;
      take x;
      now
        let i be Nat;
        assume
A8:     i in Seg n; then
A9:     x1/.i = (f/.i)`1 & x2/.i = (f/.i)`2 by A4,A7;
        i in dom x1 & i in dom x2 by A3,FINSEQ_1:def 3,A8,A6;
        hence (x`1).i = (f/.i)`1 & (x`2).i = (f/.i)`2 by A9,PARTFUN1:def 6;
      end;
      hence thesis;
  end;
