let y 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 P2Gz being PartFunc of REAL,REAL st y in J & [:[:I,J:],K:] = dom f & f is_continuous_on [:[:I,J:],K:] & f = g & P2Gz = (ProjPMap2 ((Integral2 (L-Meas,(R_EAL g))),y)) | I holds
( P2Gz || I is bounded & P2Gz is_integrable_on I )
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 P2Gz being PartFunc of REAL,REAL st y in J & [:[:I,J:],K:] = dom f & f is_continuous_on [:[:I,J:],K:] & f = g & P2Gz = (ProjPMap2 ((Integral2 (L-Meas,(R_EAL g))),y)) | I holds
( P2Gz || I is bounded & P2Gz is_integrable_on I )
let f be PartFunc of [:[:RNS_Real,RNS_Real:],RNS_Real:],RNS_Real; :: thesis: for g being PartFunc of [:[:REAL,REAL:],REAL:],REAL
for P2Gz being PartFunc of REAL,REAL st y in J & [:[:I,J:],K:] = dom f & f is_continuous_on [:[:I,J:],K:] & f = g & P2Gz = (ProjPMap2 ((Integral2 (L-Meas,(R_EAL g))),y)) | I holds
( P2Gz || I is bounded & P2Gz is_integrable_on I )
let g be PartFunc of [:[:REAL,REAL:],REAL:],REAL; :: thesis: for P2Gz being PartFunc of REAL,REAL st y in J & [:[:I,J:],K:] = dom f & f is_continuous_on [:[:I,J:],K:] & f = g & P2Gz = (ProjPMap2 ((Integral2 (L-Meas,(R_EAL g))),y)) | I holds
( P2Gz || I is bounded & P2Gz is_integrable_on I )
let P2Gz be PartFunc of REAL,REAL; :: thesis: ( y in J & [:[:I,J:],K:] = dom f & f is_continuous_on [:[:I,J:],K:] & f = g & P2Gz = (ProjPMap2 ((Integral2 (L-Meas,(R_EAL g))),y)) | I implies ( P2Gz || I is bounded & P2Gz is_integrable_on I ) )
assume that
A1: y in J and
A2: [:[:I,J:],K:] = dom f and
A3: f is_continuous_on [:[:I,J:],K:] and
A4: f = g and
A5: P2Gz = (ProjPMap2 ((Integral2 (L-Meas,(R_EAL g))),y)) | I ; :: thesis: ( P2Gz || I is bounded & P2Gz is_integrable_on I )
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;
Y-section ([:I,J:],y) = I by A1, MEASUR11:22;
then A7: ProjPMap2 (((Integral2 (L-Meas,(R_EAL g))) | [:I,J:]),y) = (ProjPMap2 ((Integral2 (L-Meas,(R_EAL g))),y)) | I 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 ( P2Gz || I is bounded & P2Gz is_integrable_on I ) by A1, A5, A6, A7, A8, MESFUN16:42; :: thesis: verum