let a, b, c, d be Real; :: thesis: for Y being RealBanachSpace
for f being continuous PartFunc of REAL, the carrier of Y st a <= b & ['a,b'] c= dom f & c in ['a,b'] & d in ['a,b'] holds
( ||.f.|| is_integrable_on ['(min (c,d)),(max (c,d))'] & ||.f.|| | ['(min (c,d)),(max (c,d))'] is bounded & ||.(integral (f,c,d)).|| <= integral (||.f.||,(min (c,d)),(max (c,d))) )
let Y be RealBanachSpace; :: thesis: for f being continuous PartFunc of REAL, the carrier of Y st a <= b & ['a,b'] c= dom f & c in ['a,b'] & d in ['a,b'] holds
( ||.f.|| is_integrable_on ['(min (c,d)),(max (c,d))'] & ||.f.|| | ['(min (c,d)),(max (c,d))'] is bounded & ||.(integral (f,c,d)).|| <= integral (||.f.||,(min (c,d)),(max (c,d))) )
let f be continuous PartFunc of REAL, the carrier of Y; :: thesis: ( a <= b & ['a,b'] c= dom f & c in ['a,b'] & d in ['a,b'] implies ( ||.f.|| is_integrable_on ['(min (c,d)),(max (c,d))'] & ||.f.|| | ['(min (c,d)),(max (c,d))'] is bounded & ||.(integral (f,c,d)).|| <= integral (||.f.||,(min (c,d)),(max (c,d))) ) )
assume A1: ( a <= b & ['a,b'] c= dom f & c in ['a,b'] & d in ['a,b'] ) ; :: thesis: ( ||.f.|| is_integrable_on ['(min (c,d)),(max (c,d))'] & ||.f.|| | ['(min (c,d)),(max (c,d))'] is bounded & ||.(integral (f,c,d)).|| <= integral (||.f.||,(min (c,d)),(max (c,d))) )

per cases ( not c <= d or c <= d ) ;

suppose A3: not c <= d ; :: thesis: ( ||.f.|| is_integrable_on ['(min (c,d)),(max (c,d))'] & ||.f.|| | ['(min (c,d)),(max (c,d))'] is bounded & ||.(integral (f,c,d)).|| <= integral (||.f.||,(min (c,d)),(max (c,d))) )

then A5: ( d = min (c,d) & c = max (c,d) ) by XXREAL_0:def 9, XXREAL_0:def 10;
then ['d,c'] c= dom f by A1, INTEGR19:3;
then integral (f,c,d) = - (integral (f,d,c)) by A3, Th1947;
then ||.(integral (f,c,d)).|| = ||.(integral (f,d,c)).|| by NORMSP_1:2;
hence ( ||.f.|| is_integrable_on ['(min (c,d)),(max (c,d))'] & ||.f.|| | ['(min (c,d)),(max (c,d))'] is bounded & ||.(integral (f,c,d)).|| <= integral (||.f.||,(min (c,d)),(max (c,d))) ) by A1, A3, A5, Lm10; :: thesis: verum

end;

suppose A6: c <= d ; :: thesis: ( ||.f.|| is_integrable_on ['(min (c,d)),(max (c,d))'] & ||.f.|| | ['(min (c,d)),(max (c,d))'] is bounded & ||.(integral (f,c,d)).|| <= integral (||.f.||,(min (c,d)),(max (c,d))) )

then ( c = min (c,d) & d = max (c,d) ) by XXREAL_0:def 9, XXREAL_0:def 10;
hence ( ||.f.|| is_integrable_on ['(min (c,d)),(max (c,d))'] & ||.f.|| | ['(min (c,d)),(max (c,d))'] is bounded & ||.(integral (f,c,d)).|| <= integral (||.f.||,(min (c,d)),(max (c,d))) ) by A1, A6, Lm10; :: thesis: verum

end;

end;