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))) | [:I,J:]),y) holds
( P2Gz is continuous & dom (ProjPMap2 (((Integral2 (L-Meas,(R_EAL g))) | [:I,J:]),y)) = I & P2Gz | I is bounded & P2Gz is_integrable_on I & integral (P2Gz,I) = Integral (L-Meas,(ProjPMap2 (((Integral2 (L-Meas,(R_EAL g))) | [:I,J:]),y))) & integral (P2Gz,I) = (Integral1 (L-Meas,((Integral2 (L-Meas,(R_EAL g))) | [:I,J:]))) . y & ProjPMap2 (((Integral2 (L-Meas,(R_EAL g))) | [:I,J:]),y) is_integrable_on L-Meas )
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))) | [:I,J:]),y) holds
( P2Gz is continuous & dom (ProjPMap2 (((Integral2 (L-Meas,(R_EAL g))) | [:I,J:]),y)) = I & P2Gz | I is bounded & P2Gz is_integrable_on I & integral (P2Gz,I) = Integral (L-Meas,(ProjPMap2 (((Integral2 (L-Meas,(R_EAL g))) | [:I,J:]),y))) & integral (P2Gz,I) = (Integral1 (L-Meas,((Integral2 (L-Meas,(R_EAL g))) | [:I,J:]))) . y & ProjPMap2 (((Integral2 (L-Meas,(R_EAL g))) | [:I,J:]),y) is_integrable_on L-Meas )
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))) | [:I,J:]),y) holds
( P2Gz is continuous & dom (ProjPMap2 (((Integral2 (L-Meas,(R_EAL g))) | [:I,J:]),y)) = I & P2Gz | I is bounded & P2Gz is_integrable_on I & integral (P2Gz,I) = Integral (L-Meas,(ProjPMap2 (((Integral2 (L-Meas,(R_EAL g))) | [:I,J:]),y))) & integral (P2Gz,I) = (Integral1 (L-Meas,((Integral2 (L-Meas,(R_EAL g))) | [:I,J:]))) . y & ProjPMap2 (((Integral2 (L-Meas,(R_EAL g))) | [:I,J:]),y) is_integrable_on L-Meas )
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))) | [:I,J:]),y) holds
( P2Gz is continuous & dom (ProjPMap2 (((Integral2 (L-Meas,(R_EAL g))) | [:I,J:]),y)) = I & P2Gz | I is bounded & P2Gz is_integrable_on I & integral (P2Gz,I) = Integral (L-Meas,(ProjPMap2 (((Integral2 (L-Meas,(R_EAL g))) | [:I,J:]),y))) & integral (P2Gz,I) = (Integral1 (L-Meas,((Integral2 (L-Meas,(R_EAL g))) | [:I,J:]))) . y & ProjPMap2 (((Integral2 (L-Meas,(R_EAL g))) | [:I,J:]),y) is_integrable_on L-Meas )
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))) | [:I,J:]),y) implies ( P2Gz is continuous & dom (ProjPMap2 (((Integral2 (L-Meas,(R_EAL g))) | [:I,J:]),y)) = I & P2Gz | I is bounded & P2Gz is_integrable_on I & integral (P2Gz,I) = Integral (L-Meas,(ProjPMap2 (((Integral2 (L-Meas,(R_EAL g))) | [:I,J:]),y))) & integral (P2Gz,I) = (Integral1 (L-Meas,((Integral2 (L-Meas,(R_EAL g))) | [:I,J:]))) . y & ProjPMap2 (((Integral2 (L-Meas,(R_EAL g))) | [:I,J:]),y) is_integrable_on L-Meas ) )
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))) | [:I,J:]),y) ; :: thesis: ( P2Gz is continuous & dom (ProjPMap2 (((Integral2 (L-Meas,(R_EAL g))) | [:I,J:]),y)) = I & P2Gz | I is bounded & P2Gz is_integrable_on I & integral (P2Gz,I) = Integral (L-Meas,(ProjPMap2 (((Integral2 (L-Meas,(R_EAL g))) | [:I,J:]),y))) & integral (P2Gz,I) = (Integral1 (L-Meas,((Integral2 (L-Meas,(R_EAL g))) | [:I,J:]))) . y & ProjPMap2 (((Integral2 (L-Meas,(R_EAL g))) | [:I,J:]),y) is_integrable_on L-Meas )
reconsider Gz = (Integral2 (L-Meas,(R_EAL g))) | [:I,J:] as PartFunc of [:REAL,REAL:],REAL by A2, A3, A4, Th32;
reconsider Fz = Gz as PartFunc of [:RNS_Real,RNS_Real:],RNS_Real ;
A6: dom (Integral2 (L-Meas,(R_EAL g))) = [:REAL,REAL:] by FUNCT_2:def 1;
A7: P2Gz = ProjPMap2 ((R_EAL Gz),y) by A5, MESFUNC5:def 7;
A8: Fz is_uniformly_continuous_on [:I,J:] by A2, A3, A4, Th34;
hence P2Gz is continuous by A6, A7, NFCONT_2:7, MESFUN16:37; :: thesis: ( dom (ProjPMap2 (((Integral2 (L-Meas,(R_EAL g))) | [:I,J:]),y)) = I & P2Gz | I is bounded & P2Gz is_integrable_on I & integral (P2Gz,I) = Integral (L-Meas,(ProjPMap2 (((Integral2 (L-Meas,(R_EAL g))) | [:I,J:]),y))) & integral (P2Gz,I) = (Integral1 (L-Meas,((Integral2 (L-Meas,(R_EAL g))) | [:I,J:]))) . y & ProjPMap2 (((Integral2 (L-Meas,(R_EAL g))) | [:I,J:]),y) is_integrable_on L-Meas )
dom (ProjPMap2 ((R_EAL Gz),y)) = I by A6, A1, A8, MESFUN16:28;
hence dom (ProjPMap2 (((Integral2 (L-Meas,(R_EAL g))) | [:I,J:]),y)) = I by MESFUNC5:def 7; :: thesis: ( P2Gz | I is bounded & P2Gz is_integrable_on I & integral (P2Gz,I) = Integral (L-Meas,(ProjPMap2 (((Integral2 (L-Meas,(R_EAL g))) | [:I,J:]),y))) & integral (P2Gz,I) = (Integral1 (L-Meas,((Integral2 (L-Meas,(R_EAL g))) | [:I,J:]))) . y & ProjPMap2 (((Integral2 (L-Meas,(R_EAL g))) | [:I,J:]),y) is_integrable_on L-Meas )
A9: Fz is_continuous_on [:I,J:] by A8, NFCONT_2:7;
hence ( P2Gz | I is bounded & P2Gz is_integrable_on I ) by A6, A7, A1, MESFUN16:42; :: thesis: ( integral (P2Gz,I) = Integral (L-Meas,(ProjPMap2 (((Integral2 (L-Meas,(R_EAL g))) | [:I,J:]),y))) & integral (P2Gz,I) = (Integral1 (L-Meas,((Integral2 (L-Meas,(R_EAL g))) | [:I,J:]))) . y & ProjPMap2 (((Integral2 (L-Meas,(R_EAL g))) | [:I,J:]),y) is_integrable_on L-Meas )
A10: ( P2Gz is_integrable_on L-Meas & integral (P2Gz,I) = Integral (L-Meas,(ProjPMap2 ((R_EAL Gz),y))) & (Integral1 (L-Meas,(R_EAL Gz))) . y = integral (P2Gz,I) ) by A6, A7, A9, A1, MESFUN16:43;
hence ( integral (P2Gz,I) = Integral (L-Meas,(ProjPMap2 (((Integral2 (L-Meas,(R_EAL g))) | [:I,J:]),y))) & integral (P2Gz,I) = (Integral1 (L-Meas,((Integral2 (L-Meas,(R_EAL g))) | [:I,J:]))) . y ) by MESFUNC5:def 7; :: thesis: ProjPMap2 (((Integral2 (L-Meas,(R_EAL g))) | [:I,J:]),y) is_integrable_on L-Meas
ProjPMap2 ((R_EAL Gz),y) is_integrable_on L-Meas by A10, A7, MESFUNC5:def 7;
hence ProjPMap2 (((Integral2 (L-Meas,(R_EAL g))) | [:I,J:]),y) is_integrable_on L-Meas by MESFUNC5:def 7; :: thesis: verum