reserve a,b,p,r,r1,r2,s,s1,s2,x0,x for Real;
reserve f,g for PartFunc of REAL,REAL;
reserve X,Y for set;

theorem Th2:
  for n being Nat, R1,R2 being Element of n-tuples_on
  REAL, r1,r2 being Real
  holds mlt(R1^<*r1*>,R2^<*r2*>)=mlt(R1,R2)^<*r1*r2*>
proof
  let n be Nat;
  let R1,R2 be Element of n-tuples_on REAL;
  let r1,r2 be Real;
   reconsider r1,r2 as Element of REAL by XREAL_0:def 1;
  len (R1^<*r1*>) = len R1 + 1 by FINSEQ_2:16
    .= n+1 by CARD_1:def 7
    .= len R2 + 1 by CARD_1:def 7
    .= len (R2^<*r2*>) by FINSEQ_2:16;
  then
A1: len(mlt(R1^<*r1*>,R2^<*r2*>)) = len (R1^<*r1*>) by FINSEQ_2:72
    .= len R1 + 1 by FINSEQ_2:16
    .= n+1 by CARD_1:def 7;
A2: for k being Nat st k in Seg(n+1) holds mlt(R1^<*r1*>,R2^<*r2*>).k = (
  mlt(R1,R2)^<*r1*r2*>).k
  proof
    let k be Nat such that
A3: k in Seg (n+1);
A4: k<=n+1 by A3,FINSEQ_1:1;
    now
      per cases by A4,XXREAL_0:1;
      suppose
        k<n+1;
        then
A5:     k<=n by NAT_1:13;
        1<=k by A3,FINSEQ_1:1;
        then
A6:     k in Seg n by A5,FINSEQ_1:1;
        then k in Seg len R1 by CARD_1:def 7;
        then k in dom R1 by FINSEQ_1:def 3;
        then
A7:     (R1^<*r1*>).k = R1.k by FINSEQ_1:def 7;
        k in Seg len R2 by A6,CARD_1:def 7;
        then k in dom R2 by FINSEQ_1:def 3;
        then
     (R2^<*r2*>).k = R2.k by FINSEQ_1:def 7;
then A8:    mlt(R1^<*r1*>,R2^<*r2*>).k = (R1.k)*(R2.k) by A7,RVSUM_1:59;
        len (mlt(R1,R2)) = n by CARD_1:def 7;
        then
    k in dom mlt(R1,R2) by A6,FINSEQ_1:def 3;
        then (mlt(R1,R2)^<*r1*r2*>).k = mlt(R1,R2).k by FINSEQ_1:def 7;
        hence thesis by A8,RVSUM_1:59;
      end;
      suppose
A9:    k=n+1;
        then k=len R1+1 by CARD_1:def 7;
        then
A10:    (R1^<*r1*>).k=r1 by FINSEQ_1:42;
A11:    n+1 = len (mlt(R1,R2)) + 1 by CARD_1:def 7;
        k=len R2+1 by A9,CARD_1:def 7;
        then
    (R2^<*r2*>).k=r2 by FINSEQ_1:42;
        then mlt(R1^<*r1*>,R2^<*r2*>).k = r1*r2 by A10,RVSUM_1:59;
        hence thesis by A9,A11,FINSEQ_1:42;
      end;
    end;
    hence thesis;
  end;
   reconsider rr = r1*r2 as Element of REAL;
  mlt(R1^<*r1*>,R2^<*r2*>) is Element of (n+1)-tuples_on REAL
     by A1,FINSEQ_2:92;
  hence thesis by A2,FINSEQ_2:119;
end;
