theorem :: NORMSP_3:67

for V, W being RealNormSpace
for L being Lipschitzian LinearOperator of V,W ex QL being Lipschitzian LinearOperator of (NVectQuot (V,(Ker L))),(Im L) ex PQL being Point of (R_NormSpace_of_BoundedLinearOperators ((NVectQuot (V,(Ker L))),(Im L))) ex PL being Point of (R_NormSpace_of_BoundedLinearOperators (V,W)) st
( QL is onto & QL is one-to-one & L = PL & QL = PQL & ||.PL.|| = ||.PQL.|| & ( for z being Point of (NVectQuot (V,(Ker L)))
for v being VECTOR of V st z = v + (Ker L) holds
QL . z = L . v ) )