let V be finite-dimensional RealUnitarySpace; :: thesis: for W being Subspace of V holds
( dim V = dim W iff (Omega). V = (Omega). W )

let W be Subspace of V; :: thesis: ( dim V = dim W iff (Omega). V = (Omega). W )
consider A being finite Subset of V such that
A1: A is Basis of V by Def1;
hereby :: thesis: ( (Omega). V = (Omega). W implies dim V = dim W )
set A = the Basis of W;
consider B being Basis of V such that
A2: the Basis of W c= B by RUSUB_3:24;
the carrier of W c= the carrier of V by RUSUB_1:def 1;
then reconsider A9 = the Basis of W as finite Subset of V by ;
reconsider B9 = B as finite Subset of V by Th3;
assume dim V = dim W ; :: thesis:
then A3: card the Basis of W = dim V by Def2
.= card B by Def2 ;
A4: now :: thesis: not the Basis of W <> B
assume the Basis of W <> B ; :: thesis: contradiction
then the Basis of W c< B by A2;
then card A9 < card B9 by CARD_2:48;
hence contradiction by A3; :: thesis: verum
end;
reconsider B = B as Subset of V ;
reconsider A = the Basis of W as Subset of W ;
(Omega). V = UNITSTR(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V, the scalar of V #) by RUSUB_1:def 3
.= Lin B by RUSUB_3:def 2
.= Lin A by
.= UNITSTR(# the carrier of W, the ZeroF of W, the addF of W, the Mult of W, the scalar of W #) by RUSUB_3:def 2
.= (Omega). W by RUSUB_1:def 3 ;
hence (Omega). V = (Omega). W ; :: thesis: verum
end;
consider B being finite Subset of W such that
A5: B is Basis of W by Def1;
A6: A is linearly-independent by ;
assume (Omega). V = (Omega). W ; :: thesis: dim V = dim W
then UNITSTR(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V, the scalar of V #) = (Omega). W by RUSUB_1:def 3
.= UNITSTR(# the carrier of W, the ZeroF of W, the addF of W, the Mult of W, the scalar of W #) by RUSUB_1:def 3 ;
then A7: Lin A = UNITSTR(# the carrier of W, the ZeroF of W, the addF of W, the Mult of W, the scalar of W #) by
.= Lin B by ;
A8: B is linearly-independent by ;
reconsider B = B as Subset of W ;
reconsider A = A as Subset of V ;
dim V = card A by
.= dim (Lin B) by A6, A7, Th9
.= card B by
.= dim W by ;
hence dim V = dim W ; :: thesis: verum