let x be Element of REAL ; :: thesis: for I, J, K being non empty closed_interval Subset of REAL
for f being PartFunc of [:[:RNS_Real,RNS_Real:],RNS_Real:],RNS_Real
for g being PartFunc of [:[:REAL,REAL:],REAL:],REAL
for P1Gz being PartFunc of REAL,REAL st x in I & [:[:I,J:],K:] = dom f & f is_continuous_on [:[:I,J:],K:] & f = g & P1Gz = (ProjPMap1 ((Integral2 (L-Meas,(R_EAL g))),x)) | J holds
( P1Gz || J is bounded & P1Gz is_integrable_on J )
let I, J, K be non empty closed_interval Subset of REAL; :: thesis: for f being PartFunc of [:[:RNS_Real,RNS_Real:],RNS_Real:],RNS_Real
for g being PartFunc of [:[:REAL,REAL:],REAL:],REAL
for P1Gz being PartFunc of REAL,REAL st x in I & [:[:I,J:],K:] = dom f & f is_continuous_on [:[:I,J:],K:] & f = g & P1Gz = (ProjPMap1 ((Integral2 (L-Meas,(R_EAL g))),x)) | J holds
( P1Gz || J is bounded & P1Gz is_integrable_on J )
let f be PartFunc of [:[:RNS_Real,RNS_Real:],RNS_Real:],RNS_Real; :: thesis: for g being PartFunc of [:[:REAL,REAL:],REAL:],REAL
for P1Gz being PartFunc of REAL,REAL st x in I & [:[:I,J:],K:] = dom f & f is_continuous_on [:[:I,J:],K:] & f = g & P1Gz = (ProjPMap1 ((Integral2 (L-Meas,(R_EAL g))),x)) | J holds
( P1Gz || J is bounded & P1Gz is_integrable_on J )
let g be PartFunc of [:[:REAL,REAL:],REAL:],REAL; :: thesis: for P1Gz being PartFunc of REAL,REAL st x in I & [:[:I,J:],K:] = dom f & f is_continuous_on [:[:I,J:],K:] & f = g & P1Gz = (ProjPMap1 ((Integral2 (L-Meas,(R_EAL g))),x)) | J holds
( P1Gz || J is bounded & P1Gz is_integrable_on J )
let P1Gz be PartFunc of REAL,REAL; :: thesis: ( x in I & [:[:I,J:],K:] = dom f & f is_continuous_on [:[:I,J:],K:] & f = g & P1Gz = (ProjPMap1 ((Integral2 (L-Meas,(R_EAL g))),x)) | J implies ( P1Gz || J is bounded & P1Gz is_integrable_on J ) )
assume that
A1: x in I and
A2: [:[:I,J:],K:] = dom f and
A3: f is_continuous_on [:[:I,J:],K:] and
A4: f = g and
A5: P1Gz = (ProjPMap1 ((Integral2 (L-Meas,(R_EAL g))),x)) | J ; :: thesis: ( P1Gz || J is bounded & P1Gz is_integrable_on J )
reconsider F2 = (Integral2 (L-Meas,(R_EAL g))) | [:I,J:] as PartFunc of [:RNS_Real,RNS_Real:],RNS_Real by A2, A3, A4, Th32;
reconsider G2 = F2 as PartFunc of [:REAL,REAL:],REAL ;
A6: dom (Integral2 (L-Meas,(R_EAL g))) = [:REAL,REAL:] by FUNCT_2:def 1;
X-section ([:I,J:],x) = J by A1, MEASUR11:22;
then A7: ProjPMap1 (((Integral2 (L-Meas,(R_EAL g))) | [:I,J:]),x) = (ProjPMap1 ((Integral2 (L-Meas,(R_EAL g))),x)) | J by MESFUN12:34;
A8: (Integral2 (L-Meas,(R_EAL g))) | [:I,J:] = R_EAL (G2 | [:I,J:]) by MESFUNC5:def 7;
F2 is_uniformly_continuous_on [:I,J:] by A2, A3, A4, Th34;
then F2 is_continuous_on [:I,J:] by NFCONT_2:7;
hence ( P1Gz || J is bounded & P1Gz is_integrable_on J ) by A1, A5, A6, A7, A8, MESFUN16:40; :: thesis: verum