let V be RealUnitarySpace; :: thesis:
let V1 be Subset of V; :: thesis:
assume that
A1: V1 <> {} and
A2: V1 is linearly-closed ; :: thesis:
reconsider D = V1 as non empty set by A1;
reconsider d1 = 0. V as Element of D by A2, RLSUB_1:4;
set S = the scalar of V || V1;
set VV = the carrier of V;
set M = the Mult of V | [:REAL ,V1:];
dom the Mult of V = [:REAL ,the carrier of V:] by FUNCT_2:def 1;
then A3: dom (the Mult of V | [:REAL ,V1:]) = [:REAL ,the carrier of V:] /\ [:REAL ,V1:] by RELAT_1:90;
[:REAL ,V1:] c= [:REAL ,the carrier of V:] by ZFMISC_1:118;
then A4: dom (the Mult of V | [:REAL ,V1:]) = [:REAL ,D:] by A3, XBOOLE_1:28;

let y be set ; :: thesis:
thus ( y in D implies ex x being set st
( x in dom (the Mult of V | [:REAL ,V1:]) & y = (the Mult of V | [:REAL ,V1:]) . x ) ) :: thesis:

assume A5: y in D ; :: thesis:
then reconsider v1 = y as Element of the carrier of V ;
A6: [1,y] in [:REAL ,D:] by A5, ZFMISC_1:106;
then (the Mult of V | [:REAL ,V1:]) . [1,y] = 1 * v1 by FUNCT_1:72
.= y by RLVECT_1:def 11 ;
hence ex x being set st
( x in dom (the Mult of V | [:REAL ,V1:]) & y = (the Mult of V | [:REAL ,V1:]) . x ) by A4, A6; :: thesis:

end;

given x being set such that A7: x in dom (the Mult of V | [:REAL ,V1:]) and
A8: y = (the Mult of V | [:REAL ,V1:]) . x ; :: thesis:
consider x1, x2 being set such that
A9: x1 in REAL and
A10: x2 in D and
A11: x = [x1,x2] by A4, A7, ZFMISC_1:def 2;
reconsider xx1 = x1 as Real by A9;
reconsider v2 = x2 as Element of the carrier of V by A10;
[x1,x2] in [:REAL ,V1:] by A9, A10, ZFMISC_1:106;
then y = xx1 * v2 by A8, A11, FUNCT_1:72;
hence y in D by A2, A10, RLSUB_1:def 1; :: thesis:

end;

then D = rng (the Mult of V | [:REAL ,V1:]) by FUNCT_1:def 5;
then reconsider M = the Mult of V | [:REAL ,V1:] as Function of [:REAL ,D:],D by A4, FUNCT_2:def 1, RELSET_1:11;
set A = the addF of V || V1;
dom the addF of V = [:the carrier of V,the carrier of V:] by FUNCT_2:def 1;
then A12: dom (the addF of V || V1) = [:the carrier of V,the carrier of V:] /\ [:V1,V1:] by RELAT_1:90;
then reconsider S = the scalar of V || V1 as Function of [:D,D:],REAL by FUNCT_2:38;
A13: dom (the addF of V || V1) = [:D,D:] by A12, XBOOLE_1:28;

let y be set ; :: thesis:
thus ( y in D implies ex x being set st
( x in dom (the addF of V || V1) & y = (the addF of V || V1) . x ) ) :: thesis:

assume A14: y in D ; :: thesis:
then reconsider v1 = y, v0 = d1 as Element of the carrier of V ;
A15: [d1,y] in [:D,D:] by A14, ZFMISC_1:106;
then (the addF of V || V1) . [d1,y] = v0 + v1 by FUNCT_1:72
.= y by RLVECT_1:10 ;
hence ex x being set st
( x in dom (the addF of V || V1) & y = (the addF of V || V1) . x ) by A13, A15; :: thesis:

end;

given x being set such that A16: x in dom (the addF of V || V1) and
A17: y = (the addF of V || V1) . x ; :: thesis:
consider x1, x2 being set such that
A18: ( x1 in D & x2 in D ) and
A19: x = [x1,x2] by A13, A16, ZFMISC_1:def 2;
reconsider v1 = x1, v2 = x2 as Element of the carrier of V by A18;
[x1,x2] in [:V1,V1:] by A18, ZFMISC_1:106;
then y = v1 + v2 by A17, A19, FUNCT_1:72;
hence y in D by A2, A18, RLSUB_1:def 1; :: thesis:

end;

then D = rng (the addF of V || V1) by FUNCT_1:def 5;
then reconsider A = the addF of V || V1 as Function of [:D,D:],D by A13, FUNCT_2:def 1, RELSET_1:11;
set W = UNITSTR(# D,d1,A,M,S #);
UNITSTR(# D,d1,A,M,S #) is Subspace of V by Th18;
hence ex W being strict Subspace of V st V1 = the carrier of W ; :: thesis: