
theorem Th14:
for a,b,c,d be Real, f be PartFunc of REAL,REAL st c <= d &
 [.c,d.] c= [.a,b.] & [.a,b.] c= dom f &
 f|[.a,b.] is bounded & f is_integrable_on ['a,b'] &
 f|[.a,b.] is nonnegative holds
  integral(f,c,d) <= integral(f,a,b)
proof
    let a,b,c,d be Real, f be PartFunc of REAL,REAL;
    assume that
A1:  c <= d and
A2:  [.c,d.] c= [.a,b.] and
A3:  [.a,b.] c= dom f and
A4:  f|[.a,b.] is bounded and
A5:  f is_integrable_on ['a,b'] and
A6:  f|[.a,b.] is nonnegative;

A7:a <= c & d <= b by A1,A2,XXREAL_1:50; then
A8: a <= d by A1,XXREAL_0:2; then
A9: a <= b by A7,XXREAL_0:2;

A10:[.a,b.] = ['a,b'] by A8,A7,XXREAL_0:2,INTEGRA5:def 3; then
    c in ['a,b'] by A1,A2,XXREAL_1:1; then
A11:integral(f,a,b) = integral(f,a,c)+integral(f,c,b)
      by A9,A3,A4,A5,A10,INTEGRA6:17;

A12:c <= b by A1,A7,XXREAL_0:2; then
    [.a,c.] c= [.a,b.] by XXREAL_1:34; then
A13:[.a,c.] c= dom f by A3;

A14:f|[.a,c.] is bounded by A4,A12,XXREAL_1:34,RFUNCT_1:74;
    (f|[.a,b.])|[.a,c.] is nonnegative by A6,MESFUNC6:55; then
    f|[.a,c.] is nonnegative by A12,XXREAL_1:34,RELAT_1:74; then
    0 <= integral(f,a,c) by A7,A13,A14,Th12; then
A15:integral(f,c,b) <= integral(f,a,b) by A11,XREAL_1:31;

A16:f is_integrable_on ['c,b'] & f|['c,b'] is bounded
      by A3,A4,A5,A10,A7,A12,INTEGRA6:18;

A17:[.c,b.] = ['c,b'] by A1,A7,XXREAL_0:2,INTEGRA5:def 3;
    [.c,b.] c= [.a,b.] by A7,XXREAL_1:34; then
A18:['c,b'] c= dom f by A3,A17;
    d in ['c,b'] by A1,A7,A17,XXREAL_1:1; then
A19:integral(f,c,b) = integral(f,c,d)+integral(f,d,b)
      by A12,A16,A18,INTEGRA6:17;

    [.d,b.] c= [.a,b.] by A8,XXREAL_1:34; then
A20:[.d,b.] c= dom f by A3;

A21:f|[.d,b.] is bounded by A4,A8,XXREAL_1:34,RFUNCT_1:74;
    (f|[.a,b.])|[.d,b.] is nonnegative by A6,MESFUNC6:55; then
    f|[.d,b.] is nonnegative by A8,XXREAL_1:34,RELAT_1:74; then
    0 <= integral(f,d,b) by A7,A20,A21,Th12; then
    integral(f,c,d) <= integral(f,c,b) by A19,XREAL_1:31;
    hence integral(f,c,d) <= integral(f,a,b) by A15,XXREAL_0:2;
end;
