reserve X for set;
reserve n,i for Element of NAT;
reserve a,b,c,d,e,r,x0 for Real;
reserve A for non empty closed_interval Subset of REAL;
reserve f,g,h for PartFunc of REAL,REAL n;
reserve E for Element of REAL n;

theorem Th30:
  a <= b & (for x be Real st x in ['a,b'] holds f.x = E) &
  ['a,b'] c= dom f & c in ['a,b'] & d in ['a,b']
  implies integral(f,c,d) = (d-c)*E
  proof
    assume A1: a<=b & (for x be Real st x in ['a,b'] holds f.x=E) &
    ['a,b'] c= dom f & c in ['a,b'] & d in ['a,b'];
A2: Seg n = dom (integral(f,c,d)) by INTEGR15:def 18;
A3: now let i; set P = proj(i,n);
      dom (P)=REAL n by FUNCT_2:def 1; then
      rng f c= dom(P); then
A4:   ['a,b'] c= dom (P*f) by A1,RELAT_1:27;
      for x be Real st x in ['a,b'] holds (P*f).x=P.E
      proof
        let x be Real;
        assume A5: x in ['a,b']; then
A6:     f.x = f/.x by A1,PARTFUN1:def 6;
        (P*f).x = P.(f/.x) by A4,A5,A6,FUNCT_1:12
        .= P.E by A6,A5,A1;
        hence thesis;
      end;
      hence integral((P*f),c,d) = (P.E)*(d-c) by A4,A1,INTEGRA6:27;
    end;
A7: now let i be Nat;
      set P = proj(i,n);
      assume
A8:   i in dom (integral(f,c,d));
      hence (integral(f,c,d)).i = integral(P*f,c,d) by A2,INTEGR15:def 18
      .= (P.E)*(d-c) by A3,A8
      .= (d-c)*(E.i) by PDIFF_1:def 1
      .= ((d-c)*E).i by RVSUM_1:44;
    end;
    len ((d-c)*E) = n by CARD_1:def 7; then
    Seg n = dom ((d-c)*E) by FINSEQ_1:def 3;
    hence thesis by A2,A7,FINSEQ_1:13;
  end;
