 reserve x for set,
         n,m,k for Nat,
         r for Real,
         V for RealLinearSpace,
         v,u,w,t for VECTOR of V,
         Av for finite Subset of V,
         Affv for finite affinely-independent Subset of V;

theorem Th1:
  for f1,f2 be real-valued FinSequence, r be Real
    holds Intervals(f1,r)^Intervals(f2,r) = Intervals(f1^f2,r)
proof
  let f1,f2 be real-valued FinSequence;
  let r be Real;
  set I1=Intervals(f1,r),I2=Intervals(f2,r),f12=f1^f2;
  set I12=Intervals(f12,r);
  A1: dom I12=dom f12 by EUCLID_9:def 3;
  A2: len f12=len f1+len f2 & len(I1^I2)=len I1+len I2 by FINSEQ_1:22;
  A3: dom I1=dom f1 by EUCLID_9:def 3;
  then A4: len I1=len f1 by FINSEQ_3:29;
  A5: dom I2=dom f2 by EUCLID_9:def 3;
  then len I2=len f2 by FINSEQ_3:29;
  then A6: dom(I1^I2)=dom I12 by A1,A4,A2,FINSEQ_3:29;
  now let i be Nat;
   assume A7: i in dom(I1^I2);
   then A8: I12.i=].f12.i-r,f12.i+r.[ by A1,A6,EUCLID_9:def 3;
   per cases by A7,FINSEQ_1:25;
   suppose A9: i in dom I1;
    then (I1^I2).i=I1.i & f12.i=f1.i by A3,FINSEQ_1:def 7;
    hence (I1^I2).i=I12.i by A3,A8,A9,EUCLID_9:def 3;
   end;
   suppose ex j be Nat st j in dom I2 & i=len I1+j;
    then consider j be Nat such that
     A10: j in dom I2 and
     A11: i=len I1+j;
    (I1^I2).i=I2.j & f12.i=f2.j by A5,A4,A10,A11,FINSEQ_1:def 7;
    hence (I1^I2).i=I12.i by A5,A8,A10,EUCLID_9:def 3;
   end;
  end;
  hence thesis by A6,FINSEQ_1:13;
end;
